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All Textbook Solutions for Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Calculate the following expression and round to two places: 2-3+4-0.60.4If the annual percentage rate is 8% and the interest is compounded monthly, what is the amount owed on a principal of 5000 after 15 years?Valentines Day According to the National Retail Federation, on Valentines Day 2015, American men planned to spend an average of 190.53 on their sweethearts, and women planned to spend only 96.58 on their heroes. What percentage of the average male expenditure was the average female expenditure?Pet Owners According to the Humane Society, in 2015, 65% of U.S. households owned at least one pet, and 42% of households who did own pets owned at least two. The U.S. Census Bureau tells us that there were 117 million households in 2015. How many households owned at least two pets? Report your answer in millions rounded to two places.A Billion Dollars A one dollar bill is 0.0043 inch thick. If you had a billion one-dollar bills and made a stack of them, how high in miles would the stack be? Remember that there are 12 inches in a foot and 5280 feet in a mile.National Debt In mid-2015 the U.S. population was about 321 million. The national debt was well over 18 trillion. In millions of dollars, the debt was 18,151,998. How much did each American owe in mid-2015? Report your answer in thousands of dollars rounded to the nearest whole number.10% Discount and 10% Tax Suppose you want to buy a great pair of designer jeans that were originally priced at 75, but were now on sale for 10% off. When you buy the jeans, you need to pay sales tax of 10% on the sales price. How much will you have to pay for the jeans?A Good Investment You have just received word that your original investment of 850 has increased in value by 13%. What is the value of your investment today?A Bad Investment You have just received word that your original investment of 720 has decreased in value by 7%. What is the value of your investment today?An Uncertain Investment Suppose you invested 1300 in the stock market two years ago. During the first year the value of the stock increased by 12%. During the second year, the value of the stock decreased by 12%. How much money is your investment worth at the end of the two-year period? Did you earn money or lose money? Note: The answer to the first question is not 1300Pay Raise You receive a raise in your hourly pay from 9.25 per hour to 9.50 per hour. What percentage increase in your pay does this represent?Heart Disease In a certain country, the number of deaths due to heart disease decreased from 235 in one year to 221 in the next year. What percentage decrease in deaths due to heart disease does this represent?Trade Discount Often retailers sell merchandise at a suggested retail price determined by the manufacturer. The trade discount is the percentage discount given to the retailer by the manufacturer. The resulting price is the retailers net cost and so is called the cost price. For example, if the suggested retail price is 100.00 and the trade discount is 45%, then the cost price is 100.00-45%100=55.00 dollars. a. If an item has a suggested retail price of 9.99 and the trade discount is 40%, what is the retailers cost price? b. If item has a cost price of 37.00 and a suggested retail price of 65.00, what trade discount was used?Series Discount This is a continuation of Exercise 11. Sometimes manufacturers give more than one discount instead of a single trade discount for example, in trading with large-volume retailers. Such a series discount is quoted as a sequence of discounts, taken one after another. Suppose a manufacturer normally gives a trade discount of 45%, but it has too much of the item in inventory and so wants to sell more. In this case the manufacturer may give all retailers another discount of 15% and may perhaps extend yet another discount of 10% to a specific retailer it wants to land as a client. In this example, the series discount would be 45%, 15%, 10%, calculated one after another, like this: For an item with a suggested retail price of 100.00, applying the first discount gives 100.00-45%100.00=55.00 dollars. The second discount of 15% is applied to the 55.00 as follows: 55.00-15%55.00=46.75 dollars. Now, the third discount gives the final cost price of 46.75-10%46.75=42.08 dollars. a. Suppose an item has a suggested retail price of 80.00 and the manufacturer is giving a series discount of 25% and 10%. What is the resulting cost price? b. Suppose an item has suggested retail price of 100.00 and the manufacturer is giving a series discount of 35%, 10%, 5%. What is the resulting cost price? c. What single trade discount would give the same cost price as a series of 35%, 10%, 5%? Note: The answer is not 50% d. Explain why we could have calculated the same answer as in part b by multiplying 100.000.650.900.95. In this case, what would the 0.65, 0.90, and 0.95 represent? Trade Discount Often retailers sell merchandise at a suggested retail price determined by the manufacturer. The trade discount is the percentage discount given to the retailer by the manufacturer. The resulting price is the retailers net cost and so is called the cost price. For example, if the suggested retail price is 100.00 and the trade discount is 45%, then the cost price is 100.00-45%100=55.00 dollars. a. If an item has a suggested retail price of 9.99 and the trade discount is 40%, what is the retailers cost price? b. If item has a cost price of 37.00 and a suggested retail price of 65.00, what trade discount was used?Present Value Present value is the amount of money that must be invested now at a given rate of interest to produce a given future value. For a one-year investment, the present value can be determined using PresentValue=FutureValue1+r, Where r is the yearly interest rate expressed as a decimal. Thus, if yearly interest rate is 8%, then 1+r=1.08. If an investment yields an yearly interest rate of 12% is available, what is the present value of an investment that will be worth 5000 at the end of 1 year? That is, how much must be invested today at 12% in order for the investment to have a value of 5000 at the end of a year?Future Value Business and finance texts refer to the value of an investment at a future time as its future value. If an investment of P dollars is compounded yearly at an interest rate of r as a decimal, then the value of the investment after t years is given by FutureValue=P1+rt. In this formula, 1+rt is known as the future value interest factor, so the formula above can be written as FutureValue=PFuturevalueinterestfactor Financial officers normally calculate this or look it up in a table a. What future value interest factor will make an investment double? b. Say you have an investment that is compounded yearly at a rate of 9%. Find the future value interest factor for a 7-year investment. c. Use the results from part b to calculate the 7-year future value if your initial investment is 5000.The Rule of 72 This is a continuation of Exercise 14. Financial advisors sometimes use a rule of thumb known as Rule of 72 to get a rough estimate of the time it takes for an investment to double in value. For an investment that is compounded yearly at an interest rate of r%, this rule says it will take about 72/r years for the investment to double. In this calculation, r is the integer interest rate rather than a decimal. Thus, if the interest rate is 8%, we would use 72/8 rather than 72/0.08. For the remainder of this exercise, we will consider an investment that is compounded yearly at an interest rate of 13%. a. According to the Rule 72, how long will it take the investment to double in value? Parts b and c of this exercise will check to see how accurate this estimate is for this particular case. b. Using the answer you got from part a of this exercise, calculate the future value interest factor as defined in Exercise 14. Is it exactly the same as your answer to the part a of Exercise 14? c. If your initial investment was 5000, use your answer from part b to calculate the future value. Did your investment exactly double? Future Value Business and finance texts refer to the value of an investment at a future time as its future value. If an investment of P dollars is compounded yearly at an interest rate of r as a decimal, then the value of the investment after t years is given by FutureValue=P1+rt. In this formula, 1+rt is known as the future value interest factor, so the formula above can be written as FutureValue=PFuturevalueinterestfactor Financial officers normally calculate this or look it up in a table a. What future value interest factor will make an investment double? b. Say you have an investment that is compounded yearly at a rate of 9%. Find the future value interest factor for a 7-year investment. c. Use the results from part b to calculate the 7-year future value if your initial investment is 5000.The Truth in Lending Act Many lending agencies compound interest more often than yearly, and, as we noted in Example P.2, they are required to report the annual percentage rate, or APR, in a prominent place on the loan agreement. Furthermore, they are required to calculate the APR in a specific way. If r is the monthly interest rate, then APR is calculated using APR=12r. a. Suppose a credit card company charges a monthly interest rate of 1.9%. What APR must the company report? b. The phrase annual percentage rate leads some people to believe that if you barrow 6000 from a credit card company that quotes an APR of 22.8%, and if no payments are made, then at the end of 1 year, the interest would be calculated as 22.8% simple interest on 6000. How much would you owe at the end of a year if interest is calculated in this way? c. If Interest is calculated monthly which is common, then the actual amount you would owe in the situation of part b is given by 60001.01912. What is the actual amount you would owe at the end of a year?The Size of the Earth The radius of the Earth is approximately 4000 miles. a. How far is it around the equator? Hint: You are looking for the circumference of a circle b. What is the volume of the Earth? Hint: The volume of a sphere of radius r is given by 43r3. b. What is the surface area of the Earth? Hint: The surface area of a sphere of radius r is given by 4r2.When the Radius Increases a. A rope is wrapped tightly around a wheel with radius of 2 feet. If the radius of the wheel is increased by 1 foot to a radius 3 feet, by how much must the rope be lengthened to fit around the wheel? b. Consider a rope wrapped around the Earths equator. We noted in Exercise 17 that the radius of the Earth is about 4000 miles. That is 21,120,000 feet. Suppose now that rope is to be suspended exactly 1 foot above the equator. By how much must the rope be lengthened to accomplish this? The Size of the Earth The radius of the Earth is approximately 4000 miles. a. How far is it around the equator? Hint: You are looking for the circumference of a circle b. What is the volume of the Earth? Hint: The volume of a sphere of radius r is given by 43r3. b. What is the surface area of the Earth? Hint: The surface area of a sphere of radius r is given by 4r2.The Length of Earths Orbit The Earth is approximately 93 million miles from the sun. For this exercise, we will assume that the Earths orbit is a circle 2. a. How far does the Earth travel in a year? b. What is the velocity in miles per year of the Earth in its orbit? Hint: Recall that Velocity =DistanceTime. c. How many hours are there in a year? Note: Assume a year is 365 days d. What is the velocity in miles per hour of the Earth in its orbit?20E21E22E23ELean Body Weight in Males A persons lean body weight L is the amount that he or she would weigh if all body fat were marginally to disappear. One text 3 gives the equation that practitioners can use most feasibly in the field to predict lean body weight in young males. The equation is L=98.42+1.08W-4.14A Here L is the lean body weight in pounds, W is the weight in pounds and A is the abdominal circumference in inches. Find the approximate lean body weight of a young adult male who weighs 188 pounds and has an abdominal circumference of 35 inches. What is the weight of his body fat? What is his body fat percent?Lean Body Weight in Females This is a continuation of Exercise 24. The text cited in Exercise 24 gives a more complex method of calculating lean body weight for young adult females: L=19.81+0.73W+21.2R-0.88A-1.39H+2.43F Here L is the lean body weight in pounds, W is the weight in pounds, R is wrist diameter, A is the abdominal circumference in inches, H is hip circumference if inches, and F is forearm circumference in inches. According to this formula, what is the approximate lean body weight of a young adult female who weighs 132 pounds and has wrist diameter of 2 inches, abdominal circumference of 27 inches, hip circumference of 37 inches and forearm circumference of 7 inches? What is the weight of her body fat? What is her body fat percent? Lean Body Weight in Males A persons lean body weight L is the amount that he or she would weigh if all body fat were marginally to disappear. One text gives the equation that practitioners can use most feasibly in the field to predict lean body weight in young males. The equation is L=98.42+1.08W-4.14A Here L is the lean body weight in pounds, W is the weight in pounds and A is the abdominal circumference in inches. Find the approximate lean body weight of a young adult male who weighs 188 pounds and has an abdominal circumference of 35 inches. What is the weight of his body fat? What is his body fat percent?Mannings Equation Hydrologists sometimes use Mannings equation to calculate the velocity v in feet per second, of water flowing through a pipe. The velocity depends on the hydraulic radius R in feet, which is one-quarter of the diameter of the pipe when the pipe if flowing full; the slope S of the pipe, which gives vertical drop in feet for each horizontal foot; and the roughness coefficient n, which depends on the material of which the pipe is made. The relationship is given by v=1.486nR2/3S1/2 For a certain brass pipe, the roughness coefficient has been measured to be n=0.012. The pipe has a diameter of 3 feet and a slope of 0.2 foot per foot. That is, the pipe drops 0.2 foot for each horizontal foot. If the pipe is flowing full, find the hydraulic radius of the pipe, and find the velocity of the water flowing through the pipe.Relativistic Length A rocket ship travelling near the speed of light appears to a stationary observer to shorten with speed. A rocket ship with a length of 200 meters will appear to a stationary observer to have a length of 2001-r2 meters, where r is the ratio of the velocity of the ship to the speed of light. What is the apparent length of the rocket ship if it is travelling at a speed that is 99% of the speed of light?Equity in a Home When you purchase a home by securing a mortgage, the total paid toward the principal is your equity in the home. Technically, the lending agency calculates your equity by subtracting the amount you still owe on your mortgage from the current value of your home, which may be higher or lower than your principal. Assume that your mortgage is for 350, 000 at a monthly rate of 0.007 as a decimal and that the term of the mortgage is 30 years. Then your equity after k monthly payments is 350, 0001.007k-11.007360-1 dollars. Calculate the equity in your home after 10 years.The Advantage Cash Card At the student union on a certain campus, you can save on food purchases by using the Advantage Cash Card. You deposit money into an Advantage Cash account and are issued a credit card the you use to purchase food. The card has several advantages: If you open your Advantage Cash account for 200 or more, a 5% bonus is added to your account balance. When you use your Advantage Cash Card, you receive 5% off the retail price of any food purchase. When you buy food with cash, you must pay a sales tax of 7.375%. With the Advantage Cash Card, you pay no sales tax. a.An item retails for 1.00. What do you pay if you use your Advantage Cash Card? b.An item retails for 1.00. What do you pay if you use cash? Round your answer to five decimal places for use in part d. c.What retail value of food will you be able to purchase if you open an Advantage Cash account for 300? Suggestion: Dont forget your 5% bonus and use the results of part a d.What retail value of food would you be able to purchase with 300 if you spend it as cash at the food court? Suggestion: Use the results of part b e.Calculate the percentage increase from your answer for part d to your answer for part c. Explain in practical terms the meaning of this percentage.1SBE2SBE3SBEBasic Calculations 7.61.79.2Parenthesis and Grouping 7.3-6.82.5+1.86SBEParentheses and Grouping 6+e+13Parentheses and Grouping -e+e9SBE10SBE11SBE12SBE13SBE14SBE15SBE16SBEArithmetic In Exercises S-16 through S-20, perform the calculation and report the answer rounded to two decimal places. For some of the calculations, you may wish to use the chain calculation facility of your calculator to help avoid errors. 23.2-13+418SBE19SBE20SBE21SBE22SBE23SBE24SBE25SBE26SBE27SBE28SBE29SBEEvaluating Formulas In Exercises S-21 through S-30, you are given a formula that you are asked to evaluate with given values for some of the variables. Report your answers rounded to 2 decimal places, except for Exercise 29, where you should round to four decimal places. Evaluate the formula AA+B using A=5 and B=6.Lending Money For a certain loan, the interest I due at the end of a loan period is given by I=Prt, where P is the principal barrowed, r is the yearly interest rate as a decimal, and t is the number of years since the money was barrowed. What interest is accrued if 3 years ago we barrowed 5000 at an interest rate of 5%?Monthly Payment For a certain instalment loan, the monthly payment M is given by M=Pr1+rt1+rt-1 where P is the original amount barrowed, r is the monthly interest rate as a decimal, and t is the number of months required to pay off the loan. What is the monthly payment if the monthly interest rate as a decimal is 0.05, the amount barrowed was 12, 000, and the loan paid off in 36 months?33SBEA Skydiver When a skydiver jumps from an airplane, his downward velocity, in feet per second, before he opens his parachute, is given by v=1761-0.834t, where t is the number of seconds that have elapsed since he jumped from the airplane. What is the velocity after 5 seconds?Future Value In certain savings scenarios, the value F of an investment after t years, the future value, is given by F=P1+rt. Here r is the yearly interest rate as a decimal, P is the amount of the original investment and t is the term of the investment. If we invest 1000 at an interest rate of 0.06 per year as a decimal, and if the term of the investment is 5 years, what is the future value?continuedA Population of Deer The number N of deer in a certain population t years after observation began is given by N=12.360.03+0.55t. What is the deer population after 10 years? Round your answer to the nearest whole number.37SBEGetting Three Sixes If we roll n fair dice, then the probability of getting exactly 3 sixes not more and not less is given by P=nn-1n-275056n. What is the probability of getting exactly 3 sixes if we roll 7 fair dice?1CR2CR3CR4CRKeplers Third Law According to Keplers third law of planetary motion, the mean distance D, in millions of miles, from a planet in our solar system to the sun is related to the time P, in years, that it takes for the planet to complete a revolution around the sun, and the relationship is D=93P2/3. It takes the planet Pluto 249 years to complete a revolution around the sun. What is the mean distance from Pluto to the sun? What is the mean distance from Earth to the sun? Give your answers to the nearest million miles.Traffic Signal Traffic engineers study how long the yellow light for a traffic signal should be. For one intersection, the number of seconds n required for a yellow light is related to the average approach speed v, in feet per second, by n=1+v30+100v. If the approach speed is 80 feet per second about 55 miles per hour, how long the yellow light be?Explain the meaning of G(4,3,2) and calculate its value.What is your monthly payment if you borrow 5000 at a monthly rate of 0.61 and pay it off in 5 years?Movie Tickets According to information provided by the National Association of Theater Owners, between 2000 and 2014 the average cost of a movie ticket in a given year was C(t)=5.40+0.22t dollars. Here t is the time in years since 2000. a. Explain in practical terms the meaning of C(9). b. Use functional notation to express the average cost of a movie ticket in 2012. c. Calculate the average cost of a movie ticket in 2012.McDonalds The formula M(t)=1.19t+13.22 gives the approximate total revenue for McDonalds, in billions of dollars, t years after 2000. The formula applies to the years 2000 through 2013. a. Explain in practical terms the meaning of M(5). b. Use functional notation to express the total revenue for 2010. c. Calculate the total revenue in 2010.Speed from Skid Marks When a car makes an emergency stop on dry pavement, it leaves skid marks on the pavement. The speed S, in miles per hour, of the car when the brakes were applied is related to the length L, in feet, of the skid mark. The relationship is S(L)=5.05L a. Use functional notation to express the speed at which the skid mark will be 60feet. Then calculate that speed. b. Explain in practical terms the meaning of S(100)Harris-Benedict Formula Your basal metabolic rate is the amount of energy in calories your body needs to function at rest. The Harris-Benedict formula is used to estimate the basal metabolic rate. There is one formula for adult males and another for adult females. In these formulas, w is your body weight in pounds, h is your height in inches, a is your age in years, M=M(w,h,a) is the basal metabolic rate for adult males, and F=(w,h,a) is the basal metabolic rate for adult females: M=66+6.3w+12.7h6.8aF=655+4.3w+4.7h4.7a Use functional notation to express your own basal metabolic rate, and then calculate its value.Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. There is a formula that estimates how much your puppy will weigh when it reaches adulthood. The method we present applies to medium-sized breeds. First, find your puppys weight w, in pounds, at an age of a weeks, where a is 16 weeks or less. Then the predicted adult weight W=W(a,w) in pounds, is given by the formula. W=52wa a. Use functional notation to express the adult weight of a puppy that weighs 6 pounds at 14 weeks. b. Calculate the predicted adult weight for the puppy from part a.Gross Profit Margin The gross profit margin is a measurement of a companys manufacturing and distribution efficiency during the production process. If G is the gross profit and T is the total revenue, both in dollars, then the gross profit margin M=M(G,T) is given by the formula M=GT a. Use functional notation to express the gross profit margin for a company that has a gross profit of 335, 000 and a total revenue of 540, 000. b. Calculate the gross profit margin in part a. The gross profit margin is often expressed as a percent. Give your answer as both a decimal and a percent. c. If the gross profit stays the same but total revenue increases, would the gross profit margin increase or decrease?Tax Owed The income tax T owed in a certain state is a function of the taxable income I, both measured in dollars. The formula is T=0.11I500 a. Express using functional notation the tax owed on a taxable income of 13, 000, and then calculate that value. b. If your taxable income increases from 13,000 to 14,000, by how much does your tax increase? c. If your taxable income increases from 14,000 to 15,000, by how much does your tax increase?Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. The height of the winning pole vault in the early years of the modern Olympic Games can be modeled as a function of time by the formula H=0.05t+3.3 Here t is the number of years since 1900, and H is the winning height in meters. One meter is 39.37 inches. a. Calculate H(4) and explain in practical terms what your answer means. b. By how much did the height of the winning pole vault increase from 1900 to 1904? From 1904 to 1908?Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. V, in feet per second, is a function of the time t, in seconds, since the ball was thrown. The formula is V=4032t if we ignore air resistance. The function V is positive when the ball is rising and negative when the ball is falling. a. Express using functional notation the velocity 1 second after the ball is thrown, and then calculate that value. Is the ball rising or falling then? b. Find the velocity 2 seconds after the ball is thrown. Is the ball rising or falling then? c. What is happening 1.25 seconds after the ball is thrown? d. By how much does the velocity change from 1 to 2 seconds after the ball is thrown? From 2 to 3 seconds? From 3 to 4 seconds? Compare the answers to these three questions and explain in practical terms.Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. 10.Flushing Chlorine City water, which is slightly chlorinated, is being used to flush a tank of heavily chlorinated water. The concentration C=C(t) of chlorine in the tank t hours after flushing begins is given by C=0.1+2.78e0.37tmilligramspergallon a. What is the initial concentration of chlorine in the tank? b. Express the concentration of chlorine in the tank after 3 hours using functional notation, and then calculate its value.Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. A Population of Deer When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first, but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number N of deer present at time 1 measured in years since the herd was introduced on a certain wildlife reserve has been determined by ecologists to be given by the function N=12.360.03+0.55t Figure1 a. How many deer were initially on the reserve? b. Calculate N(10) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10th to the 15th year?Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. 12. A Car That Gets 32 Miles per Gallon The cost C of operating a certain car that gets 32 miles per gallon is a function of the price g, in dollars per gallon, of gasoline and the distance d, in miles, that you drive. The formula for C=C(g,d) is C=gd/32 dollars. a. Use functional notation to express the cost of operation if gasoline costs 98 cents per gallon and you drive 230 miles. Calculate the cost. b. Calculate C(3.53,172) and explain the meaning of the number you have calculated.Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. 13. Radioactive Substances change form over time. For example, carbon 14, which is important for radiocarbon dating, changes through radiation into nitrogen. If we start with 5 grams of carbon 14, then the amount C=C(t) of carbon 14 remaining after t years is given by C=50.5t/5730 a. Express the amount of carbon 14 left after 800 years in functional notation, and then calculate its value. b. How long will it take before half of the carbon 14 is gone? Explain how you got your answer. Hint: You might use trial and error to solve this, or you might solve it by looking carefully at the exponent..Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. A Roast is taken from the refrigerator where it had been for several days and placed immediately in a preheated oven to cook. The temperature R=R(t) of the roast t minutes after being placed in the oven is given by R=325280e0.005tdegreesFahrenheit a. What is the temperature of the refrigerator? b. Express the temperature of the roast 30 minutes after being put in the oven in functional notation, and then calculate its value. c. By how much did the temperature of the roast increase during the first 10 minutes of cooking? d. By how much did the temperature of the roast increase from the first hour to 10 minutes after the first hour of cooking?Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. What if Interest is Compounded More Often than Monthly?Some lending institutions compound interest daily or even continuously. The term continuous compounding is used when interest is being added as often as possible-that is, at each instant in time. The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part 1 of Example 1.2, we showed that if you borrow 7800 from an institution that compounds monthly at a monthly interest rate of 0.67 for an APR of 8.04 , then in order to pay off the note in 48months, you have to make a monthly payment of 190.57. a.Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why. b.Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning. c.When interest is compounded continuously, you can calculate your monthly payment M=M(P,r,t) in dollars, for a loan of Pdollars to be paid off over t months using M=P(er1)1ert, where r=APR/12 if the APR is written in the decimal form. Use this formula to calculate the monthly payment on a loan of 7800 to be paid off over 48months with an APR of 8.04. How does this answer compare the result in Example 1.2?16ERound all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. 17. How Much Can I Borrow? The function in Example 1.2 can be rearranged to show the amount of money P=P(M,r,t) in dollars, that you can afford to borrow at a monthly interest rate of r as a decimal if you are able to make t monthly payments of M dollars: P=M1r(11(1+r)t) Suppose you can afford to pay 350 per month for 4 years. a. How much money can you afford to borrow for the purchase of a car if the prevailing monthly interest rate is 0.75? That is a 9 APR. Express the answer in functional notation, and then calculate it. b. Suppose your car dealer can arrange a special monthly interest rate of 0.25 or a 3 APR. How much can you afford to borrow now? c. Even at a 3 APR, you find yourself looking at a car you can't afford, and you consider extending the period during which you are willing to make payments to 5 years. How much can you afford to borrow under these conditions?18E19E20E21EReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Sound Pressure and DecibelsSounds exerts a pressure P on the human ear. This pressure increases as the loudness of the sound increases. If the loudness D is measure in decibels and the pressure P is measured in dynes per square centimeter, then the relationship is give by P=0.00021.122D. a.Ordinary conversation has a loudness of about 65 decibels. What is the pressure exerted on the human ear by ordinary conversation? b.A decibel level of 120 causes pain to the ear and can result in damage. What is the corresponding pressure level on the ear?ReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Mitscherlichs EquationAn important agriculture problem is to determine how a quantity of nutrient, such as nitrogen, affects the growth of plants. We consider the situation wherein sufficient quantities of all but one nutrient are present. One boule of a nutrient is the amount needed to produce 50 of maximum possible yield. In 1990, E.A. Mitscherlichs proposed the following relation, which is known as Mitscherlichs Equation: Y=10.5b. Here b is the number of baules of nutrient applied, and Y is the percentage as a decimal of maximum yield produced. a.Verify that the formula predicts that 50 of maximum yield will be produced if 1 baule of nutrient is applied. b.Use functional notation to express the percentage of maximum yield produced by 3 baule of nutrient, and calculate the value. c.The exact value of a baule depends on the nutrient in question. For nitrogen, 1 baule is 223 pounds per acre. What percentage of maximum yield will be produced if 500 pounds of nitrogen per acre is present?ReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Yield Response to Several Growth FactorsThis is a continuation of Exercise 23. If more than one nutrient is considered, the formula for percentage of maximum yield is a bit more complex. For three nutrients, the formula is Y(b,c,d)=(10.5b)(10.5c)(10.5d). a.Express using functional notation the percentage of maximum yield produced from 1 baule of the first nutrient, 2 baules of the third nutrient, and 3 baules of the third nutrient, and then calculate that value. b.One baule of nitrogen is 223 pounds per acre, 1 baule of phosphorus is 45 per acre, and 1 baule of potassium is 76 pounds per acre. What percentage of maximum yield will be obtained from 200 pounds of nitrogen per acre, 100 pounds of phosphorus per acre, and 150 pounds of potassium per acre? 23. Mitscherlichs EquationAn important agriculture problem is to determine how a quantity of nutrient, such as nitrogen, affects the growth of plants. We consider the situation wherein sufficient quantities of all but one nutrient are present. One boule of a nutrient is the amount needed to produce 50 of maximum possible yield. In 1990, E.A. Mitscherlichs proposed the following relation, which is known as Mitscherlichs Equation: Y=10.5b. Here b is the number of baules of nutrient applied, and Y is the percentage as a decimal of maximum yield produced. a.Verify that the formula predicts that 50 of maximum yield will be produced if 1 baule of nutrient is applied. b.Use functional notation to express the percentage of maximum yield produced by 3 baule of nutrient, and calculate the value. c.The exact value of a baule depends on the nutrient in question. For nitrogen, 1 baule is 223 pounds per acre. What percentage of maximum yield will be produced if 500 pounds of nitrogen per acre is present?25EReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Reynolds NumberThe Reynolds number is very important in such fields as fluid flow and aerodynamics. In the case of a fluid flowing through a pipe, the Reynoldss number R is given by R=vdD. Here v is the velocity of the fluid in meters per second, d is the diameter of the pipe in meters, D is the density of the fluid in kilograms per cubic meter, and is the viscosity of the fluid measured in Newton-seconds per square meter. Generally, when the Reynolds number is above 2000, the flow becomes turbulent, and rapid mixing occurs. When the Reynolds number is less than 2000, the flow is streamline. Consider a fluid flowing through a pipe diameter 0.05 meter at a velocity of 0.2 meter per second. a.If the fluid in the pipe is toluene, its viscosity is 0.00059 newton-seconds per square meter, and its density is 867 kilograms per cubic meter. Is the flow turbulent or streamline? b.If the toluene is replaced by glycerol, then the viscosity is 1.49 newton-seconds per square meter, and the density is 1216.3 kilograms per cubic meter. Is the glycerol flow turbulent or streamline?27E28E29E30EReminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. Adjustable Rate Mortgage-Exact PaymentsThis is a continuation of Exercise 30. In Exercise 30, we only approximated the increased payment when the rate for an ARM increases: We assumed that over the first 24 months, you accrue 0 equity in your home. Your equity in a home is the total you have paid toward the principal.We address that point here. We make use of the equity formula E=P(1+r)k1(1+r)t1. Here E is the equity, in dollars, after k monthly payments. The quantities P,t, and r are defined as in Exercise 30. Round r to five decimal places. We assume as in Exercise 30 that you borrowed 325,000 at an initial APR of 4.5 with a term of 30 years. a. What equity have you accrued after 24 months? b. When your rate adjusts to 7 after 24 months, the new amount borrowed is 325,000 less your equity. The term is now 28 years. What is your new monthly payment? 30. Adjustable Rate Mortgage-Approximating PaymentsAn adjustable rate mortgage, or ARM, is a mortgage whose interest rate varies over the life of the loan. The interest rate is often tied in same fashion to the prime rate, which may go up or down. One advantage of an ARM is that it usually has an initial rate that is lower than that of a fixed rate mortgage. In the summer 2007, defaults on home mortgages led to a crisis in the U.S. economy. At least part of the blame was placed on ARMs. This exercise illustrates the difficulties that many homeowners faced during this period. We make use of the following formula for the monthly payment: M=Pr(1+r)t(1+r)t1. Here M is the monthly payment, in dollars, P is the amount borrowed, in dollars, t is the term of the loan, in months, and r is the monthly interest rate as a decimal, with r=APR/12. In this exercise, round r to five decimal places. Suppose you purchased a home in 2005, securing a mortgage of 325,000 with a 30-year ARM. a. In 2005, interest rates were at a historical lows. Suppose that at the time of the loan, the rate for your ARM was a 4.5 APR. Calculate your monthly payment. b. Suppose you earn 6000 per month. What percentage of your income is going toward your house payment? c. Suppose that after 24 payments, your ARM rate adjusted to 7 APR. We will assume that after 24 months, your loan balance is still 325,000. This is not as unreasonable an assumption as it may appear. The correct calculation is shown in Exercise 31. What is your monthly payment now? Be careful: The term of the loan is now 28 years, not 30 years. d. Using the assumptions of part c, what percentage of your income is going to your house payment now?32E1SBE2SBE3SBE4SBE5SBE6SBEEvaluating Formulas In Exercises S-1 through S-24, evaluate the given function as required. Calculate h(3,2.2,9.7) if h(x,y,z)=xy/z8SBE9SBE10SBE11SBE12SBE13SBE14SBE15SBEEvaluating Formulas In Exercises S-1 through S-24, evaluate the given function as required. Calculate N(0)and N(10) if N(t)=12.360.03+0.55t.17SBE18SBE19SBE20SBE21SBE22SBE23SBE24SBE25SBEWhat Formulas Mean In Exercises S-25 through S-33, you are asked to relate functional notation to practical explanations of what certain functions mean. Grocery Bill If c(p,s,h) is the cost of buying p bags of potato chips, ssodas, and h hot dogs, use functional notation to express the cost of buying 2 bags of potato chips, 3 sodas, and 5 hot dogs.27SBEWhat Formulas Mean In Exercises S-25 through S-33, you are asked to relate functional notation to practical explanations of what certain functions mean. Time to Pay Off a Loan The function T(P,r,m) gives the time required to pay off an installment loan of P dollars if the APR is r as a decimal and we make monthly payments of in dollars. Use functional notation to indicate the time required to pay off a 12,000 automobile loan at an APR of 5 if the monthly payment is 450.29SBE30SBE31SBE32SBEWhat Formulas Mean In Exercises S-25 through S-33, you are asked to relate functional notation to practical explanations of what certain functions mean. Dow Jones Industrial Average The function D(t) gives the Dow Jones Industrial Average t minutes after noon today. Explain in practical terms the meaning of D(120).TEST YOUR UNDERSTANDING|FOR EXAMPLE 1.3 Use the notion of the example. Explain the meaning of W(1973), and use the average rate of change to estimate its value. Example 1.3 WOMEN EMPLOYED OUTSIDE THE HOME Table 1.2 shows the number W of women in the United States employed outside the home as a function of the date d. It was taken from the Statistical Abstract of the United States TABLE 1.2 Number of Women Employed Outside the Home in the United States d=Year 1970 1980 1990 2000 2010 W=Numberinmillions 31.5 45.5 56.8 66.3 71.9 Part 1 Explain the meaning of W(1970) and give its value. Part 2 Explain the meaning of W(1975) and estimate its value. Part 3 Express the number of women employed outside the home in 1972 in functional notation, and use the average yearly rate of change from 1970 to 1980 to estimate its value. Part 4 According to the 1950 Statistical Abstract of the United States, in 1943 there were 18.7 million women employed outside the home, and in 1946 the number was 16.8 million. On the basis of this information, find the average yearly rate of change in W from 1943 to 1946 and the average decrease per year over this time interval. Use the result to estimate W(1945).Use the table of values you made in part 4 of the example to find the limiting value of the average rate of change in velocity.Minimum Wage The table below is taken from the website of the U.S. Department of Labor. It shows the minimum wage for each decade from 1950 to 2010. The figures are adjusted for inflation and expressed in constant 2012 dollars. y=Year m=Minimumwage 1950 7.01 1960 7.59 1970 9.28 1980 8.46 1990 6.66 2000 6.90 2010 7.67 a. Find the value of m(1990). b. Use functional notation to express the minimum wage in 1985. c. Use the table to estimate the minimum wage in 1985. The actual value was 7.09.Mortgage Rates The following table is taken from the website of Freddie Mac. It shows rates for 30-year fixed-rate mortgages since 1970. y=Year r=Mortgagerate 1975 9.05 1980 13.74 1985 12.43 1990 10.13 1995 7.93 2000 8.05 2005 5.87 2010 4.69 2015 3.84 a. Explain in practical terms the meaning of r(2003). b. Use the table to estimate the value of r(2003).Box Office Hits The table below shows the highest grossing movie of the given year. The amount is the domestic box office gross, in millions of dollars. Year Movie Amount(millions) 2006 Pirates of the Caribbean: Dead Mans chest 423.32 2007 Spider-Man 3 336.53 2008 The Dark Knight 533.35 2009 Avatar 760.51 2010 Toy Story 3 415.00 2011 Harry Potter and the Deathly Hallows: Part 2 381.01 2012 The Avengers 623.28 2013 The Hunger Games: Catching Fire 424.67 2014 American Sniper 350.13 Let M=M(y) denote the highest grossing movie in year y, and let B=B(y) denote the gross for that movie. a. Give the values of M(2014) and B(2014). b. Use the functional notation to indicate the amount for the movie with the highest gross in 2013.Mobile Phone Sales In 2000, mobile handset sales totaled 414.99million. In 2005, the total was 778.75million. Let M=M(t) denote total mobile handset sales in year t. What was the average rate of change per year in M(t) from 2000 to 2005? Be sure to include proper units with your answer.Reminder Round all answers to two decimal places unless otherwise indicated. Choosing a Bat A chart from Dicks sporting Goods gives the recommended bat length B in inches for a man weighing between 161 and 170 pounds as a function of his height h in inches. The table is partially reproduced on the next page. h=Height B=Batlength 4548 30 4952 31 5356 31 5760 32 6164 32 6568 33 6972 33 73+ 33 a. Explain in practical terms the meaning of B(55) and give its value. b. Use functional notation to express the recommended bat length for a man weighing between 161 and 170 pounds if his height is 63inches.Freight on Class I Railroads According to the Association of American Railroads, Class I freight railroads are the line-haul freight railroads with 2006 operating revenue in excess of 346.8million. Let F=F(t) denote the freight revenue in billions of dollars of Class I railroads in year t. In 2005, Class I railroads had a freight revenue of 44.5billion. In 2007, the revenue was 52.9 billion. Calculate the average rate of change per year in F from 2005 to 2007 and explain its meaning in practical terms.The American Food Dollar The following table shows the percentage P=P(d) of the American food dollar that was spent on eating away from home at restaurants, for example as function of the date d. d=Year P=Precentspentawayfromhome 1969 25 1989 30 2009 34 a. Find P(1989) and explain what it means. b. What does P(1999) mean? Estimate its value. c. What is the average rate of change per year in percentage of the food dollar spent away from home for the period from 1989 to 2009? d. What does P(2004) mean? Estimate its value. Hint: Your calculation in part c should be useful. e. Predict the value of P(2014) and explain how you made your estimate.Gross Domestic Product The following table shows the U.S gross domestic product GDP G, in trillions of dollars, as a function of the year t. t=Year 2004 2010 2014 G=GDP(trillionofdollars) 11.87 14.66 17.42 a. Explain in practical terms what G(2004) means, and find its value. b. Use functional notation to express the gross domestic product in 2007, and estimate that value. c. What is the average yearly rate of change in G from 2010 to 2014? d. Use your answer to part c to predict the gross domestic product in the year 2022.Reminder Round all answers to two decimal places unless otherwise indicated. Internet Access The following table gives the number I=I(t) in millions, of adult Americans with Internet access in year t. t=Year 2000 2003 2009 I=MillionsofAmericans 113 166 196 a. Find I(2000) and explain what it means. b. Find the average rate of change per year during the period from 2003 to 2009. c. Estimate the value of I(2007). Explain how you got your answer.Reminder Round all answers to two decimal places unless otherwise indicated. A Cold Front At 4P.M. on a winter day, an arctic air mass moved from Kansas into Oklahoma, causing temperatures to plummet. The temperature T=T(h) in degrees Fahrenheit h hours after 4P.M. in Stillwater, Oklahoma, on that day is recorded in the following table. h=Hourssince4P.M. T=Temperature 0 62 1 59 2 38 3 26 4 22 a. Use functional notation to express the temperature in Stillwater at 5:30P.M., and then estimate its value. b. What was the average rate of change per minute in temperature between 5P.M. and 6P.M.? What was the average decrease per minute over that time interval? c. Estimate the temperature at 5:12P.M. d. At about what time did the temperature reach the freezing point? Explain your reasoning.A Troublesome Snowball One winter afternoon, unbeknownst to his mom, a child bring a snowball into the house, lays it on the floor, and then goes to watch T.V. Let W=W(t) be the volume of dirty water that has soaked into the carpet t minutes after the snowball was deposited on the floor. Explain in practical terms what the limiting value of W represents, and tell what has happened physically when this limiting value is reached.Reminder Round all answers to two decimal places unless otherwise indicated. Falling with a Parachute If an average-sized man jumps from an airplane with a properly opening parachute, his downward velocity v=v(t), in feet per second, t seconds into the fall is given by the following table. t=Secondsintothefall v=Velocity 0 0 1 16 2 19.2 3 19.84 4 19.97 a. Explain why you expect v to have a limiting value and what this limiting value represents physically. b. Estimate the terminal velocity of the parachutist.Reminder Round all answers to two decimal places unless otherwise indicated. t is measured in thousands of years, and C=C(t) is the amount, in grams, of carbon-14 remaining. Carbon-14 unstable radioactive t=Thousandofyears C=Gramsremaining 0 5 5 2.73 10 1.49 15 0.81 20 0.44 a. What is the average yearly rate of change of carbon-14 during the first 5000 years? b. How many grams of carbon-14 would you expect to find remaining after 1236 years? c. What would you expect to be the limiting value of C?Reminder Round all answers to two decimal places unless otherwise indicated. Newtons Law of Cooling says that a hot object cools rapidly when the difference between its temperature and that of the surrounding air is large, but the object cools more slowly when it nears room temperature. Suppose a piece of aluminum is removed from an oven and left to cool. The following table gives the temperature A=A(t), in degrees Fahrenheit. of the aluminum t minutes after it is removed from the own. a.Explain the meaning of A(75) and estimate its value. b.Find the average decrease of temperature pet minute during the first half-hour of cooling. t=Minutes A=Temperature 0 302 30 152 60 100 90 81 120 75 150 73 180 72 210 72 c.Find the average decrease per minute of temperature during the first half of the second hour of cooling. d.Explain how parts b and c support Newtons law of cooling. e.Use functional notation to express the temperature of the aluminum after I hour and 13 minutes. Estimate the temperature at that time. Note: Your work in part c should be helpful. f.What is the temperature of the oven? Exams your answer using functional notation, and give its value. g.Explain why you would expect the function A to have a limiting value. h.What is roan temperature? Explain your reasoning.15EReminder Round all answers to two decimal places unless otherwise indicated. New Construction The following table shows the value B, in billions of dollars, of new construction put in place in the United States during the year t. t=Year B=Value billions of dollars 2000 831.1 2003 891.5 2006 1167.6 2009 935.6 a. Make a table showing, for each of the 3-year periods, the average yearly rate of change in B. b. Explain in practical terms what B(2008) means, and estimate its value. c. Over what period was the growth in value of new construction the greatest? d. According to the table, in what year was the value of new construction the greatest?Reminder Round all answers to two decimal places unless otherwise indicated. Growth in Height The following table gives, for a certain man, his height H=H(t) in inches at age t in years. t=Age(years) H=Height inches 0 21.5 5 42.5 10 55.0 15 67.0 20 73.5 25 74.0 a. Use functional notation to express the height of the man at age 13, and then estimate its value. b. Now we study the mans growth rate. i. Make a table showing, for each of the 5-year periods, the average yearly growth rate-that is, the average yearly rate of change in H. ii. During which 5-year period did the man grow the most in height? iii. Describe the general trend in the mans growth rate. c. What limiting value would you estimate for the height of this man? Explain your reasoning in physical terms.Reminder Round all answers to two decimal places unless otherwise indicated. Growth in Weight The following table gives, for a certain man, his weight W=W(t) in pounds at age t in years. t=Age(years) W=Weight pounds 4 36 8 54 12 81 16 128 20 156 24 163 a. Make a table showing, for each of the 4- year periods, the average yearly rate of change in W. b. Describe in general terms how the mans gain in weight varied over time. During which 4-year period did the man gain the most in weight? c. Estimate how much the man weighed at age 30. d. Use the average rate of change to estimate how much he weighed at birth. Is your answer reasonable?Reminder Round all answers to two decimal places unless otherwise indicated. Tax Owed The following table shows the income tax T owed in a certain state as a function of the taxable income I, both measured in dollars. I=Taxableincome T=Taxowed 16,000 870 16,200 888 16,400 906 16,600 924 a. Make a table showing, for each of the intervals in the tax table above, the average rate of change in T. b. Describe the general trend in the average rate of change. What does this mean in practical terms? c. Would you expect T to have a limiting value? Be sure to explain your reasoning.Reminder Round all answers to two decimal places unless otherwise indicated. Sales Income The following table shows the net monthly income N for a real estate agency as a function of the monthly real estate sales s, both measured in dollars. s=Sales N=Netincome 450,000 4000 500,000 5500 550,000 7000 600,000 8500 a. Make a table showing, for each of the intervals in the tax table above, the average rate of change in N. What pattern do you see? b. Use the average rate of change to estimate the net monthly income for monthly real estate sales of 520,000. In light of your answer to part a, how confident are you that your estimate is an accurate representation of the actual income? c. Would you expect N to have a limiting value? Be sure to explain your reasoning.Reminder Round all answers to two decimal places unless otherwise indicated. Yellowfin Tuna Data were collected comparing the weight W, in pounds, of a yellowfin tuna to its length L, in centimeters. These data are presented in the following table. L=Length W=Weight 70 14.3 80 21.5 90 30.8 100 42.5 110 56.8 120 74.1 130 94.7 140 119 160 179 180 256 a. What is the average rate of change, in weight per centimeter of length, in going from a length of 100 centimeters to a length of 110 centimeters? b. What is the average rate of change, in weight per centimeter of length, in going from 160 to 180 centimeters? c. Judging from the data in the table, does an extra centimeter of length make more difference in weight for a small tuna or for a large tuna? d. Use the average rate of change to estimate the weight of a yellowtuna fish that is 167 centimeters long? e. What is the average rate of change, in length per pound of weight, in going from a weight of 179 pounds to a weight of 256 pounds? f. What would you expect to be the length of a yellow tuna weighing 225 pounds?Reminder Round all answers to two decimal places unless otherwise indicated. Arterial Blood Flow Medical evidence shows that a small change in the radius of an artery can indicate a large change in blood flow. For example, if one artery has a radius only 5 larger than another, the blood flow rate is 1.22 times as large. Further information is given in the table below. Increase in radius Times greater blood flow rate 5 1.22 10 1.46 15 1.75 20 2.07 a. Use the average rate of change to estimate how many times greater the blood flow rate is in an artery that has a radius 12 larger than another. b. Explain why if the radius is increased by 12 and then we increase the radius of the new artery by 12 again, the total increase in the radius is 25.44. c. Use parts a and b to answer the following question: How many times greater is the blood flow rate in an artery that 25.44 larger in radius than another? d. Answer the question in part c using the average rate of change.23ERound all answers to two decimal places unless otherwise indicated. 24. Timber Stumpage Price The following table shows timber stumpage prices for pine pulpwood in two regions of the American South. Prices are in dollars per ton and were recorded at the start of the indicated year. Year Southeast Mid-Atlantic 2002 6.50 4.00 2005 7.50 6.70 2007 7.00 10.00 a. For the Mid-Atlantic, what is the average rate of change per year in price from 2002 to 2005? b. Use your answers to part a to estimate the price in the Mid-Atlantic region at the start of 2004. The actual price was 6.40 per ton. c. For the southeast, what is the average rate of change per year in price form 2005 to 2007? d. Use your answer to part c to estimate the price in the Southeast at the start of 2008. e. For each region, find the percentage increase in price from 2002 to 2007. f. On the basis of your answer to part e, in the absence of other factors, would an investor in timber be better advised to choose the Southeast or the Mid-Atlantic?Round all answers to two decimal places unless otherwise indicated. 25. The Margaria-Kalamen Test The Margaria Kalamen test is used by physical educators as a measure of leg strength. An individual runs up a staircase, and the elapsed time from the third to the ninth step is recorded. The power score is calculated using a formula involving the individuals weight, the height of the stairs, and the running time. The following table shows the power scores required of men for an excellent rating for selected ages. Age Power score for excellent rating 18 224 25 210 35 168 45 125 55 98 a. What is the average rate of change per year in excellence level from 25years to 35years old? b. What power score would merit an excellent rating for a 27-year-old man? c. During which 10-years from age 25 to 55 would you expect to see the greatest decrease in leg power?26E27EReminder Round all answers to two decimal places unless otherwise indicated. Defense SpendingData about recent federal defense spending are given in the accompanying Statistical Abstract of the United States table. Here t denotes the time, in years, since 1990 and D denotes federal defense spending, in billions of dollars. a.Calculate the average yearly rate of change in defense spending from 1990 to 1995. b.Use your answer from part a to estimate D(3), and explain what it means. t= Years since 1990 D= Spending billions of dollars 0 328.4 5 310.0 10 341.5 15 565.5 20 843.8 c.Calculate the average yearly rate of change in defense spending from 2005 to 2010. d.Use your answer form part c to estimate the value of D(22).1SBE2SBEFor these exercises, round all estimates to one decimal place. A Tabulated Function The following table gives values for a function N=N(t). t N=N(t) 10 17.6 20 23.8 30 44.6 40 51.3 50 53.2 60 53.7 70 53.9 Exercises S-1 through S-20 refer to this function. Function Values Find the value of N(30).4SBEFor these exercises, round all estimates to one decimal place. A Tabulated Function The following table gives values for a function N=N(t). t N=N(t) 10 17.6 20 23.8 30 44.6 40 51.3 50 53.2 60 53.7 70 53.9 Exercises S-1 through S-20 refer to this function. Function Values Find the value of N(50).6SBE7SBE8SBE9SBE10SBE11SBE12SBEFor these exercises, round all estimates to one decimal place. A Tabulated Function The following table gives values for a function N=N(t). t N=N(t) 10 17.6 20 23.8 30 44.6 40 51.3 50 53.2 60 53.7 70 53.9 Exercises S-1 through S-20 refer to this function. Averaging Use averaging to estimate the value of N(65).14SBE15SBE16SBE17SBE18SBE19SBEFor these exercises, round all estimates to one decimal place. A Tabulated Function The following table gives values for a function N=N(t). t N=N(t) 10 17.6 20 23.8 30 44.6 40 51.3 50 53.2 60 53.7 70 53.9 Exercises S-1 through S-20 refer to this function. Limiting Values Assuming that the function N describes a physical situation for which a limiting value is expected, estimate the limiting value of N.21SBE22SBE23SBE24SBE25SBE26SBE27SBE28SBEFor these exercises, round all estimates to one decimal place. Another Table The following is a partial table of values for f=f(x). x f=f(x) 0 5.7 5 4.3 10 1.1 15 3.6 20 7.9 Use this table to complete Exercises S-21 through S-30. Average Rate of Change Calculate the average rate of change from x=15 to x=20. Use your answer to estimate the value of f(19).For these exercises, round all estimates to one decimal place. Another Table The following is a partial table of values for f=f(x). x f=f(x) 0 5.7 5 4.3 10 1.1 15 3.6 20 7.9 Use this table to complete Exercises S-21 through S-30. Average Rate of Change Use the results from Exercise S-29 to estimate the value of f(25).When Limiting Values Occur Suppose S(t) represents the average speed, in miles per hour, for a 100-mile trip that requires t hours. Explain why we expect S to have a limiting value.Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it continues on its course indefinitely. Let D(t) denote its distance from Earth after t years of travel. Do you expect that D has a limiting value?TEST YOUR UNDERSTANDING | FOR EXAMPLE 1.5 From 2000 through 2011, what was the lowest value the euro attained? When did that happen? EXAMPLE 1.5 CURRENCY EXCHANGE Figure 1.11 shows the value E=E(d), in U.S. dollars, of the euro as a function of the date d. We have added grid lines to the graph to aid in reading function values from the graph. Part 1 Explain the meaning of F2003 and estimate its value. Part 2 From 2000 through 2011, what was the largest value the euro attained? When did that happen? Part 3 What was the average yearly increase in the value of the euro from 2006 to 2009? Part 4 During which one-year period was the graph increasing most rapidly? Part 5 As an American investor, would you have made money if you bought euros in 2002 and sold them in 2008?TEST YOUR UNDERSTANDING | FOR EXAMPLE 1.6 Locate a region where the graph is increasing and concave down. Explain what is happening to the value of the euro during the period you select. EXAMPLE 1.6 CONCAVITY AND CURRENCY EXCHANGE Consider once more the value of the euro as given by the graph in Figure 1.11. Part 1 From 2000 to late 2003, is the graph concave up or concave down? Part 2 Explain in practical terms what the concavity means about the value of the euro during this period.3TU4TUReminderRound all answers to two decimal places unless otherwise indicated. Sketching a Graph with Given Concavity: a.Sketch a graph that is always decreasing but starts out concave down and then changes to concave up. There should be a point of inflection in your picture. Mark and label it. b.Sketch a graph that is always decreasing but starts out concave up and then changes to concave down. There should be a point of inflection in your picture. Mark and label it.ReminderRound all answers to two decimal places unless otherwise indicated. An InvestmentIn 2010, an investor put money into a fund. The graph in Figure 1.26 shows the value, v=v(d) of the investment, in dollars, as a function of date d. a.Express the original investment using functional notation and give its value. b.Is the graph concave up or concave down? Explain what this means about the growth in value of the account. c.When will the value of the investment reach 55,000? FIGURE 1.26 An investment d.What is the average yearly increase from 2050 to 2060? e.Which is larger, the average yearly increase from 2050 to 2060 or the average yearly increase from 2010 to 2020? Explain your reasoning.ReminderRound all answers to two decimal places unless otherwise indicated. Household DebtThe graph in Figure 1.27 shows the average American household debt h as a function of the date d. Here h(d) represents house hold debt as a percentage of disposable income. a.Explain the meaning of h(1975) in practical terms, and use the graph to find its value. b.The graph reaches a maximum value. Estimate the maximum value and the year in which it occurred. FIGURE 1.27Reminder Round all answers to two decimal places unless otherwise indicated. Auto Loan Rates The graph in Figure 1.28 shows rates r for new car loans as a function of the date d. Explain in general terms the behavior of r(d) from 1980 to 2010.5EReminderRound all answers to two decimal places unless otherwise indicated. Equilibrium PriceThis is a continuation of Exorcise 5. The equilibrium price is the price where the supply and demand are the same. In Figure 1.31, the supply curve is in red and the demand curve is in blue. Use this graph to estimate the equilibrium price. How many items are supplied at the equilibrium price? FIGURE 1.31 5. Supply and Demand CurvesA supply curve is a graph that shows the quantity of a product that is made available by suppliers as a function of the price. Similarly, a demand curve is a graph that shows the quantity of a product that consumers are willing to purchase as a function of the price. Examples of supply and demand curves are shown in Figures 1.29 and 1.30. a.Explain in practical terms what the supply curve in Figure 1.29 tells us. b.Explain in practical terms what the demand curve in Figure 1.30 tells us. FIGURE 1.29 A Supply curve FIGURE 1.30 A demand curveReminderRound all answers to two decimal places unless otherwise indicated. Skirt LengthOne authors analyzed skirt lengths in the United Kingdom UK and Germany. To get a standardized measure of skirt length, she calculated the ratio. LengthfromshouldertoskirthemLengthfromshouldertoankle The article includes the following graph of the ratio for skirt length in terms of the year. In the graph, the solid line is the ratio in the UK, and the dashed line is the ratio in Germany. FIGURE 1.32 Ratio for skirt length a.Does a larger ratio indicate a longer or a shorter skirt? b.When were skirt lengths shortest in the UK? c.In this period, did skirt lengths in either Germany or the UK ever reach to the ankle?8E9E10E11E12EReminderRound all answers to two decimal places unless otherwise indicated. River FlowThe graph in Figure 1.37 shows the mean flow F for the Arkansas River, in cubic feet of water per second, as a function of the time t, in months, since the start of the year. The flow is measured near the rivers headwaters in the Rocky Mountains. a.Use functional notation to express the flow at the end of July, and then estimate that value. b.When is the flow at its greatest? c.At what time is the flow increasing the fastest? FIGURE 1.37 Flow for the Arkansas River d.Estimate the average rate of change per month in the flow during the first 2 months of the year. e.In light of the source of the Arkansas River, interpret your answers to parts b, c, and d.14EReminderRound all answers to two decimal places unless otherwise indicated. Cutting TreesIn forestry management, it is important to know the net stumpage value of a stand that is, a group of trees. This is the commercial value of the trees minus the costs of felling, hauling, etc. The graph in Figure 1.39 shows the net stumpage value V, in dollars per acre, of a Douglas fir stand in the Pacific Northwest as a function of the age t, in years, of the stand. FIGURE 1.39 Net stumpage value of a Douglas Fir a. Estimate the net stumpage value of a Douglas fir 1 stand that is 60 years old. b. Estimate the age of a Douglas fir stand whose net stumpage value is 40,000peracre. c. At what age does the commercial value of the stand equal the costs of felling, hauling, etc.? d. At what age is the net stumpage value increasing the fastest? e. This graph shows V only up to age t=160years, but the Douglas fir lives for hundreds of years. Draw a graph to represent what you expect for V over the life span of the tree. Explain your reasoning.ReminderRound all answers to two decimal places unless otherwise indicated. Wind ChillThe graph in Figure 1.40 shows the temperature T = Tv adjusted for wind chill as a function of the velocity v of the wind when the thermometer reads 30 degrees Fahrenheit. The adjusted temperature T shows the temperature that has an equivalent cooling power when there is no wind. a. At what wind speed is the temperature adjusted for wind chill equal to 0? b. Your answer in part a is the solution of an equation involving Tv. Which equation? c. At what value of v would a small increase in v have the greatest effect on Tv? In other words, at what wind speed could you expect a small increase in wind speed to cause the greatest change in wind chill? Explain your reasoning. d. Suppose the wind speed is 45 miles per hour. Judging from the shape of the graph, how significant would you expect the effect on Tv to be if the wind speed increased? FIGURE 1.40 Temperature adjusted for wind chill when the thermometer reads 30 degrees Fahrenheit.ReminderRound all answers to two decimal places unless otherwise indicated. Tornadoes In OklahomaThe graph in Figure 1.41 shows the number T=T(d) of tornadoes in Oklahoma Fl/EF1 and stronger reported by the National Climatic Data Center. Round your answers to the nearest whole number. a. When were the most tornadoes reported? How many were reported in that year? b. When were the fewest tornadoes reported? How many were reported in that year? c. What was the average yearly rate of decrease in tornadic activity from 2011 to 2012? d. What was the average yearly rate of increase in tornadic activity from 2009 to 2011? e. What was the average yearly rate of change that is, the average yearly rate of total increase or total decrease in tornadic activity from 2009 to 2012? FIGURE 1.41 Tornadoes in OklahomaReminderRound all answers to two decimal places unless otherwise indicated. InflationDuring a period of high inflation, a political leader was up for re-election. Inflation had been increasing during his administration, but he announced that the rate of increase of inflation was decreasing. Draw a graph of inflation versus time that illustrates this situation. Would this announcement convince you that economic conditions were improving?19E20E21EReminderRound all answers to two decimal places unless otherwise indicated. Protein Content of Wheat GrainProtein content of wheat grain is affected by soil moisture and the amount of available nitrogen among other things. Figure 1.45 shows" the percent of protein content of wheat grain versus pounds of nitrogen per acre applied in three separate situations. In each case, soil moisture refers to moisture at the soil depth of 2 inches to 12 inches. Situation 1: Irrigation was used when soil moisture dropped to 49. Situation 2: Irrigation was used when soil moisture dropped to 34. Situation 3: Irrigation was used when soil moisture dropped to 1. a. If irrigation begins when soil moisture reaches 49, what application of nitrogen will result in the lowest percentage of protein in wheat grain? b. If irrigation begins when soil moisture reaches 34, what application of nitrogen will result in the same protein content of wheat grain as beginning irrigation when soil moisture reaches 1? c. If you irrigate when soil moisture reaches 34, how much nitrogen should you apply to achieve a 13 protein content in wheat grain? d. Does Figure 1.45 indicate that, for nitrogen levels at 45 pounds per acre or higher, increased protein content in wheat grain is associated with higher or lower soil moisture? FIGURE 1.45 Protein content versus availability of nitrogen23EReminderRound all answers to two decimal places unless otherwise indicated. HydrographsWhen a rainfall brings more water than the soil can absorb, runoff occurs, and hydrologists refer to the event as a rainfall excess. The easiest way to envision runoff is to think of a watershed that drains into the mouth of a single stream. The runoff is the number of cubic feet per minute cfpm being dumped into the mouth of the stream. An important way of depicting runoff is the hydrograph, which is simply the graph of total discharge, in cubic feet per minute, versus time. A typical runoff hydrograph is shown in Figure 1.47. The horizontal axis is hours since rainfall excess began. A hydrograph displays a number of important features. a. Time to peak is the elapsed time from the start of rainfall excess to peak runoff. What is the time to peak shown by the hydrograph in Figure 1.47? b. Time of concentration is the elapsed time from the end of rainfall excess to the inflection point after peak runoff. The end of rainfall excess is not readily apparent from a hydrograph, but it occurs before the peak. If the end of rainfall excess occurred 5 hours after the start of rainfall excess. estimate the time of concentration from Figure 1.47. c. Recession time is the time from peak runoff to the end of runoff. Estimate the recession time for the hydrograph in Figure 1.47. d. Time base is the time from beginning to end of surface runoff. What is the time base for the hydrograph in Figure 1.47? FIGURE 1.47 A runoff hydrograph25E26E27EReminder Round all answers to two decimal places unless otherwise indicated. Relativistic Time A correctly functioning hourglass on board a rocket ship would appear to a stationary observer to take longer than an hour to empty. The graph in Figure 1.51 shows the time T in hours, that it appears to a stationary observer to take an hour-glass to empty if it is on a rocket ship moving at a given percentage r, as a decimal, of the speed of light. a. Use the graph to estimate the value of T(0.6) and explain in practical terms what it means. b. Is the graph concave up or concave down? c. Does a small change in velocity cause a greater change in apparent time at slower speeds or at faster speeds? FIGURE 1.50 Relativistic length of an hour29E30E31E1SBE2SBE3SBE4SBE5SBE6SBE7SBE8SBE9SBE10SBE11SBE12SBE13SBE14SBE15SBE16SBE17SBEA Function Given by a Graph The following is the graph of a function f=f(x). Exercises S-15 through S-23 refer to this graph. S-18. What is the minimum value of f(x)?19SBE20SBE21SBE22SBE23SBE24SBE25SBE26SBE27SBE28SBE29SBE30SBE31SBE32SBE33SBE34SBE35SBETEST YOUR UNDERSTANDING Suppose we start with 9 milligrams of pollutant per liter of water, as in the example. Assume that the cleaning process decreases the concentration of pollutant by 30 each hour. What is the pollutant concentration after each of the first 3 hours? EXAMPLE 1.9 PURIFYING WATER Water that is initially contaminated with a concentration of 9 milligrams of pollutant per liter of water is subjected to a cleaning process. The cleaning process is able to reduce the pollutant concentration by 25 each hour. Let C=C(t) denote the concentration, in milligrams per liter, of pollutant in the water r hours after the purification process begins. Part 1 What is the concentration of pollutant in the water after 3 hours? Part2 Compare the results predicted by the verbal description at the end of each of the first 3 hours with those given by the formula C=90.75t. Part 3 The formula given in part 2 is, in fact, the correct one. Use it to find the concentration of pollutant after 414 hours of cleaning.2TU3TU4TU1E2EReminder: Round all answer to two decimal places unless otherwise indicated. 3.Fat in Fast Food You are shopping for dinner for your family in a fast-food restaurant, but you are concerned about fat. An informative menu gives you the following information. Item Hamburger Chicken Sandwich Fries Onion rings Grams of fat 24 13 30 25 a. How much total fat is in an order of 3 hamburgers, 1 chicken sandwich, 2 orders of fries, and 2 orders of onion rings? b. Use a formula to express the total grams of fat F in an order of h hamburgers, c chicken sandwiches, f orders of fries, and o orders of onion rings.Reminder: Round all answer to two decimal places unless otherwise indicated. 4.Thanksgiving Dinner It takes 10 minutes to preheat your oven to 325 degrees. For an oven preheated to 325 degrees, the recommended cooking time for a turkey is about 15 minutes per pound. a. Use a formula to express the total time T, in minutes, needed to preheat the oven and then bake a turkey weighing p pounds. b. Use your formula from part a to find the approximate time required to prepare a turkey weighing 18 pounds.5. United States Population Growth In 1960 the population of the United States was about 180 million. Since that time, the population has increased by approximately 1.2 each year. This is a verbal description of the function N=N(t), where N is the population, in millions, and t is the number of years since 1960. a. Express in functional notation the population of the United States in 1963. Calculate its value. b. Use the verbal description of N to make a table of values that shows U.S. population in millions from 1960 through 1965. c. Make a graph of U.S. population versus time. Be sure to label your graph appropriately. d. Verify that the formula 1801.012tmillion people, where t is the number of years since 1960, gives the same values as those you found in the table in part b. Note: Because t is the number of years since 1960, you would use t = 2 to get the population in 1962. e. Assuming that the population has been growing at the same percentage rate since 1960, what value does the formula above give for the population in 2000? Note: The actual population in 2000 was about 281 million..6Ea. Use a formula to express the altitude of a helicopter as a function of time. Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the altitude of the helicopter 90 seconds after takeoff, and then calculate that value. c. Make a graph of altitude versus time covering the first 3 minutes of the flight. Explain how the description of the function is reflected in the shape of the graph.Reminder: Round all answer to two decimal places unless otherwise indicated. a. Use a formula to express the world record time as a function of the time since 1950. Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the world record time in the year 1955, and then calculate that value. c. Would you expect the formula to be valid indefinitely? Be sure to explain your answer.Reminder: Round all answer to two decimal places unless otherwise indicated. a. Calculate the rental charge if you rent a car for 2 days and drive 100 miles. b. Use a formula to express the cost of renting a car as a function of the number of days you keep it and the number of miles you drive. Identify the function and each variable you use, and state the units. c. It is about 250 miles from Dallas to Austin. Use functional notation to express the cost to rent a car in Dallas, drive it to Austin, and return it in Dallas 1 week later. Use the formula from part b to calculate the cost.Reminder: Round all answer to two decimal places unless otherwise indicated. a. How much does it cost to prepare and mail a 3-page letter if your secretary spends 2 hours on typing and corrections? b. Use a formula to express the cost of preparing and mailing a letter as a function of the number of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. c. Use the function you made in part b to find the cost of preparing and mailing a 2-page letter that it takes your secretary 25 minutes to type. Note: 25 minutes is 25/60 hour.Remainder: Round all answer to two decimal places unless otherwise indicated. a. How much does the stationery alone cost for a 3-page letter? b. How much does it cost to prepare and mail a 3-page letter if your secretary spends 2 hours on typing and corrections? c. Use a formula to express the cost of the stationery alone for a letter as a function of the number of pages in the letter. Identify the function and each of the variables you use, and state the units. d. Use a formula to express the cost of preparing and mailing a letter as a function of the number of pages in the letter and the time it takes your secretary to type it. Identify the function and each of the variables you use, and state the units. e. Use the function you made in part d to find the cost of preparing and mailing a 2-page letter that it takes your secretary 25 minutes to type.m Miles per Gallon The cost of operating a car depends on the gas mileage in that your car gets, the cost x per gallon of gasoline, and the distance d that you drive. a. How much does it cost to drive 100 miles if your car gets 25 miles per gallon and gasoline costs 349 cents per gallon? b. Find a formula that gives the cost C as a function of m, g, and d be sure to state the units of each variable. c. Use functional notation to show the cost of driving a car that gets 28 miles per gallon a distance of 138 miles if gasoline costs 3.69 per gallon Use the formula from part b to calculate the cost.13EContinued This is a continuation of Exercise 13. As we saw earlier, the stock turnover rate of an item is the number of times that the average inventory of the item needs to be replaced as a result of sales in a given time period. Suppose that a hardware store sells 80 shovels each year. a. Suppose that the hardware store maintains an average inventory of 5 shovels. What is the annual stock turnover rate for the shovels? How is this related to the yearly number of orders to the wholesaler needed to restock inventory? b. What would he the annual stock turnover rate if the store maintained an average inventory of 20 shovels? c. Write a formula expressing the annual stock turnover rate as a function of the average inventory of shovels, identify the function and the variable, and state the units.Reminder: Round all answer to two decimal places unless otherwise indicated. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To deter mine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost C of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.Reminder Round all answers to two decimal places unless otherwise indicated. Total Revenue and ProfitThis is a continuation of Exercise 15. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at a break-even point. Suppose the manufacturer of widgets in Exercise 15 sells the widgets for 25each. a.Use a formula to express this manufacturers total revenue R in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b.Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use. c.Express using functional notation the profit of this manufacturer if there are 250 widgets produced in a month, and then calculate that value. d.At the production level of 250 widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of 1000 widgets per month? 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.Reminder Round all answers to two decimal places unless otherwise indicated. More on RevenueThis is a continuation of Exercises 15 and 16. In general, the highest price p per unit of an item which a manufacturer can sell N items is not constant, but is rather a function of N. The total revenue R is still the product of p and N, but the formula for R is more complicated when p depends on N. Suppose the manufacturer of widgets in Exercises 15 and Exercises 16 no longer sells widgets for 25 each. Rather, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. a.Verify that the formula p=500.01N where p is the price in dollars, give the same values as those in the table. N=Numberofwidgetssold p=Price 100 49 200 48 300 47 400 46 500 45 b.Use the formula from part a and the fact that R is the product of p and N to find a formula expressing the total revenue R as a function of N for this widget manufacturer. c.Express using functional notation the total revenue of this manufacturer if there are 450 weights produced in a month, and then calculate that value. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value. 16.Total Revenue and ProfitThis is a continuation of Exercise 15. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at break-even point. Suppose the manufacturer of widgets in Exercise 15 sells the widgets for 25each. a.Use a formula to express this manufacturers total revenue R in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b.Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use. c.Express using functional notation the profit of this manufacturer if there are 250 widgets produced in a month, and then calculate that value. d.At the production level of 250 widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of 1000 widgets per month?Reminder Round all answers to two decimal places unless otherwise indicated. More on ProfitThis is a continuation of Exercises 15, 16, and 17. In this exercise, we use the formula for the total cost of the widget manufacturer found in Exercise 15 and the formula for the total revenue found in Exercise 17. a.Use a formula to express the profit P of this manufacturer as a function of N. b.Consider the three production levels: N=200, N=700, and N=1200 . For each of these, determine whether the manufacturer has a loss, turns a profit, or is a break even point. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value. 16.Total Revenue and ProfitThis is a continuation of Exercise 15. The total revenue R for a manufacturer during a given time period is a function of the number N of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at break-even point. Suppose the manufacturer of widgets in Exercise 15 sells the widgets for 25each. a.Use a formula to express this manufacturers total revenue R in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b.Use a formula to express the profit P of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use. c.Express using functional notation the profit of this manufacturer if there are 250 widgets produced in a month, and then calculate that value. d.At the production level of 250 widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of 1000 widgets per month? 17.More on RevenueThis is a continuation of Exercise 15 and 16. In general, the highest price p per unit of an item which a manufacturer can sell N items is not constant, but is rather a function of N. The total revenue R is still the product of p and N, but the formula for R is more complicated when p depends on N. Suppose the manufacturer of widgets in Exercises 15 and Exercises 16 no longer sells widgets for 25 each. Rather, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. a.Verify that the formula p=500.01N where p is the price in dollars, give the same values as those in the table. N=Numberofwidgetssold p=Price 100 49 200 48 300 47 400 46 500 45 b.Use the formula from part a and the fact that R is the product of p and N to find a formula expressing the total revenue R as a function of N for this widget manufacturer. c.Express using functional notation the total revenue of this manufacturer if there are 450 weights produced in a month, and then calculate that value.Reminder Round all answers to two decimal places unless otherwise indicated. Renting Motel Rooms You own a motel with 30 rooms and have a pricing structure that encourages rentals of rooms in groups. One room rents for 85.00, two for 83.00 each, and in general the group rate per room is found by taking 2 off the base of 85 for each extra room rented. a.How much money do you charge per room if a group rents 3 rooms? What is the total amount of money you take in? b.Use a formula to give the rate you charge for each room if you rent n rooms to an organization. c.Find a formula for a function R=R(n) that gives the total revenue from renting n rooms to a convention host. d.Use functional notation to show the total revenue from renting a block of 9 rooms to a group. Calculate that value.Reminder:-Round all answers to two decimal places unless otherwise indicated. A Cattle Pen A rancher wants to use a fence as an enclosure for a rectangular cattle pen with area 400 square feet. a. Suppose he decides to make one side of the pen 40 feet long. Draw a picture and label the length of each side of the pen. What is the length of each side? What is the total amount of fence needed? b. What would be the total amount of fence needed if the pen were a square? c. Can two rectangles with the same area have different perimeters? d. Find a formula for a function F=F(l) that gives the total amount of fence, in feet, required in terms of the length l, in feet, of one of its sides. Hint: First draw a picture of the pen and label one side 1. Next figure out the lengths of the other sides in terms of l.Reminder Round all answers to two decimal places unless otherwise indicated. Catering a Dinner You are having a dinner catered. You pay a rental fee of 150 for the dining hall, and you pay the caterer 10 for each person who attends the dinner. a. Suppose you just want to break even. i.How much should you charge per ticket if you expect 50 people to attend? iiUse a formula to express the amount you should charge per ticket as a function of the number of people attending. Be sure to explain the meaning of the letters you choose and the units. iii.You expect 65 people to attend the dinner. Use your answer to part ii to express in functional notation the amount you should charge per ticket, and then calculate that amount. b. Suppose now that you want to make a profit of 100 from the dinner. Use a formula to express the amount you should charge per ticket as a function of the number of people attending. Again, be sure to explain the meaning of the letters you choose and the units.22E