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All Textbook Solutions for Finite Mathematics and Applied Calculus (MindTap Course List)
55E56E57E58E59E60EAverage Profit The FeatureRich Software Company sells its graphing program, Dogwood, with a volume discount. If a customer buys x copies, then he or she pays 500x. It cost the company $10,000 to develop the pro- gram and $2 to manufacture each copy. If a single customer were to buy all the copies of Dogwood, how many copies would the customer have to buy for FeatureRich Softwares average profit per copy to be maximized? How are average profit and marginal profit related at this number of copies?62E63EResource Allocation Your company is the largest sock manufacturer in the solar system, and production is automated through the use of androids and robots. Daily operating costs amount to w 200 per android and w 8 per robot.19 To meet production deadlines, the company calculates that the numbers of androids and robots must satisfy the constraint xy=1,000,000, where x is the number of androids and y is the number of robots. Assuming that the company wishes to meet production deadlines at a minimum cost, how many androids and how many robots should it use? [HINT: See Example 5.]65E66E67EPrison Population The prison population in the United States can be approximated by N(t)=0.02t32t2+100t+1,100thousandpeople(0t18). (t is the year since 1990.) When, to the nearest year, was the prison population increasing most rapidly? When was it increasing least rapidly? [HINT: You are being asked to find the extreme values of the rate of change of the prison population.]69E70E71E72E73E74E75EAgriculture Two years ago, your orange orchard contained 50 trees, and the yield per tree was 75 bags of oranges. Last year, you removed 10 of the trees and noticed that the yield per tree increased to 80 bags. Assuming that the yield per tree depends linearly on the number of trees in the orchard, what should you do this year to maximize your total yield?77E78E79E80E81E82E83E84E85EExplain how you would solve an optimization problem of the following form. Maximize P=f(x,y,z) subjectto z=g(x,y)andy=h(x).In Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=3x262EIn Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=2x4EIn Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=4x0.4x6EIn Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=e(x1)xIn Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=ex+exIn Exercises 1-10, calculate d2ydx2. [HINT: See Quick Example 1-3.] y=1xlnx10EIn Exercises 1116 the position s of a point (in feet) is given as a function of time t (in seconds). Find (a) its acceleration as a function of t and (b) its acceleration at the specified time. [HINT: See Example 1.] s=12+3t16t2;t=2In Exercises 1116 the position s of a point (in feet) is given as a function of time t (in seconds). Find (a) its acceleration as a function of t and (b) its acceleration at the specified time. [HINT: See Example 1.] s=12+t16t2;t=213EIn Exercises 1116 the position s of a point (in feet) is given as a function of time t (in seconds). Find (a) its acceleration as a function of t and (b) its acceleration at the specified time. [HINT: See Example 1.] s=1t1t2;t=2In Exercises 1116 the position s of a point (in feet) is given as a function of time t (in seconds). Find (a) its acceleration as a function of t and (b) its acceleration at the specified time. [HINT: See Example 1.] s=t+t2;t=416E17E18E19E20E21E22EIn Exercises 1724 the graph of a function is given. Find the approximate coordinates of all points of inflection of each function (if any). [HINT: See Quick Examples 5 and 6.]24E25E26E27E28E29E30E31E32E33EIn Exercises 3344, find the x-coordinates of all critical points of the given function. Determine whether each critical point is a relative maximum, a relative minimum, or neither, by first applying the second derivative test, and, if the test fails, by some other method. [HINT: See Quick Examples 79.] f(x)=2x22x+335E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54EAcceleration on Mars If a stone is dropped from a height of 40 meters above the Martian surface, its height in meters after t seconds is given by s=401.9t2. What is its acceleration? [HINT: See Example 1.]56EMotion in a Straight Line The position of a particle moving in a straight line is given by s=t3t2 feetafter t seconds. Find an expression for its acceleration after a time t. Is its velocity increasing or decreasing when t=1?Motion in a Straight Line The position of a particle moving in a straight line is given by s=3et8t2 feetafter t seconds. Find an expression for its acceleration after a time t. Is its velocity increasing or decreasing when t=1?Bottled Water Sales Annual sales of bottled water in the United States in the period 20072014 could be approximated by R(t)=0.08t20.26t+8.8billiongallons(0t7), where t is time in years since 2007. According to the model, were annual sales of bottled water accelerating or decelerating in 2011? How fast? [HINT: See Example 2.]Bottled Water Sales Annual U.S. per capita sales of bottled water in the period 20072014 could be approximated by R(t)=0.25t2t+29gallons(0t7), where t is time in years since 2007. According to the model, were annual U.S. per capita sales of bottled water accelerating or decelerating in 2009? How fast? [HINT: See Example 2.]Embryo Development The daily oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird the oxygen consumption can be approximated by c(t)=0.065t3+3.4t222t+3.6millilitersperday(8t30), where t is the time (in days) since the egg was laid. (An egg will typically hatch at around t=28.) Use the model to estimate the following (give the units of measurement for each answer and round all answers to two significant digits): a. The daily oxygen consumption 20 days after the egg was laid b. The rate at which the oxygen consumption is changing 20 days after the egg was laid c. The rate at which the oxygen consumption is accelerating 20 days after the egg was laidEmbryo Development The daily oxygen consumption of a turkey embryo increases from the time the egg is laid through the time the chick hatches. In a brush turkey the oxygen consumption can be approximated by c(t)=0.028t3+2.9t244t+95millilitersperday(20t50), where t is the time (in days) since the egg was laid (An egg will typically hatch at around t=50.) Use the model to estimate the following (give the units of measurement for each answer and round all answers to two significant digits): a. The daily oxygen consumption 40 days after the egg was laid b. The rate at which the oxygen consumption is changing 40 days after the egg was laid c. The rate at which the oxygen consumption is accelerating 40 days after the egg was laidInflation The following graph shows the approximate value of the United States Consumer Price Index (CPI) from December 2006 through July 2007: I(t) CPI Dec 2006-July 2007 The approximating curve shown on the figure is given by I(t)=0.04t3+0.4t2+0.1t+202(0t7), where t is time in months since the start of December 2006. a. Use the model to estimate the monthly inflation rate in February 2007 (t=2). [Recall that the inflation rate is I(t)/I(t).] b. Was inflation slowing or speeding up in February 2007? c. When was inflation speeding up? When was inflation slowing? [HINT: See Example 3.]64EInflation The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from July 2005 through March 2006: I(t) CPI July 2005-Mar 2006 The approximating curve shown on the figure is given by I(t)=0.06t30.8t2+3.1t+195(0t8), where t is time in months since the start of July 2005. a. Use the model to estimate the monthly inflation rate in December 2005 and February (t=5andt=7). b. Was inflation slowing or speeding up in February 2006? c. When was inflation speeding up? When was inflation slowing? [HINT: See Example 3.]Inflation The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from March 2006 through May 2007. I(t) CPI Mar 2006-May 2007 The approximating curve shown on the figure is given by I(t)=0.02t30.38t2+2t+200(0t14), where t is time in months since the start of July 2005. a. Use the model to estimate the monthly inflation rate in September 2006 and January 2007 (t=6andt=10). b. Was inflation slowing or speeding up in January 2007? c. When was inflation speeding up? When was inflation slowing? [HINT: See Example 3.]67EScientific Research: 19832003 The percentage of research articles in the prominent journal Physical Review that were written by researchers in Europe during 19832003 can be modeled by P(t)=7.01+5.4(1.2)t, where t is time in years since 1983. The graphs of P,P,andP are shown here: Determine, to the nearest whole number, the values of t for which the graph of P is concave up and where it is concave down, and locate any points of inflection. What does the point of inflection tell you about science articles? [HINT: See Example 4.]69E70E71E72EEpidemics The following graph shows the total number n of people (in millions) infected in an epidemic as a function of time t (in years): Year a. When, to the nearest year, was the rate of new infection largest? b. When could the Centers for Disease Control and Prevention announce that the rate of new infection was beginning to drop? [HINT: See Example 4.]74E75E76E77E78E79E80EOil Imports from Mexico Daily oil production in Mexico and daily U.S. oil imports from Mexico during 20092013 can be approximated by P(t)=3.10.014tmillionbarrels(9t13)I(t)=1.70.063tmillionbarrels(9t13), wheret is time in years since the start of 2000. Graph the function I(t)/P(t) andits derivative. Is the graph of I(t)/P(t) concaveup or concave down? The concavity of I(t)/P(t) tellsyou that (A) the percentage of oil produced in Mexico that was exported to the United States was decreasing. (B) the percentage of oil produced in Mexico that was not exported to the United States was increasing. (C) the percentage of oil produced in Mexico that was exported to the United States was decreasing at a slower rate. (D) the percentage of oil produced in Mexico that was exported to the United States was decreasing at a faster rate.Oil Imports from Mexico Repeat Exercises 81 using instead models for 2000-2004 shown below: P(t)=3.0+0.13tmillionbarrels(9t4)I(t)=1.4+0.06tmillionbarrels(9t4), (where t is time in years since the start of 2000.)83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98EIn Exercises 126, sketch the graph of the given function, indicating (a) x- and y-intercepts, (b) extrema, (c) points of inflection, (d) behavior near singular points of f, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. [HINT: See Example 1.] f(x)=x2+2x+1In Exercises 126, sketch the graph of the given function, indicating (a) x- and y-intercepts, (b) extrema, (c) points of inflection, (d) behavior near singular points of f, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. [HINT: See Example 1.] f(x)=x22x1In Exercises 126, sketch the graph of the given function, indicating (a) x- and y-intercepts, (b) extrema, (c) points of inflection, (d) behavior near singular points of f, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. [HINT: See Example 1.] g(x)=x312x,domain[4,4]4E5E6E7E8E9E10E11E12EIn Exercises 126, sketch the graph of the given function, indicating (a) x- and y-intercepts, (b) extrema, (c) points of inflection, (d) behavior near singular points of f, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. [HINT: See Example 1.] f(x)=x+1x14E15E16E17E18E19EIn Exercises 126, sketch the graph of the given function, indicating (a) x- and y-intercepts, (b) extrema, (c) points of inflection, (d) behavior near singular points of f, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. [HINT: See Example 1.] k(x)=2x5(x2)2/5 [Use technology for x-intercepts.] [HINT: See Example 2.]21E22E23E24E25E26E27E28E29E30E31E32E33E34EConsumer Price Index The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from July 2005 through March 2006: I(t) CPI July 2005-Mar 2006 The approximating curve shown on the figure is given by I(t)=0.06t30.8t2+3.1t+195(0t8), where t is time in months. (t=0 representsJuly 2005.) a. Locate the intercepts, extrema, and points of inflection of the curve, and interpret each feature in terms of the CPI. (Approximate all coordinates to one decimal place.) [HINT: See Example 1.] b. Recall from Section 12.2 that the inflation rate is defined to be I(t)I(t). What do the stationary extrema of the curve shown above tell you about the inflation rate?36E37E38EAverage Cost: iPhone Assume that is costs Apple approximately C(x)=400,000+160x+0.001x2 dollars to manufacture x 32GB iPhone 6s in an hour at the Foxconn Technology Group. Obtain the average cost function, sketch its graph, and analyze the graphs important features. Interpret each feature in terms of iPhone 6s. [HINT: Recall that the average cost function is C(x)=c(x)/x.]Average Cost: PlayStation 4s Assume that it costs sony approximately C(x)=800,000+340x+0.0005x2 dollars to manufacture x PlayStation 4s in an hour. Obtain the average cost function, sketch its graph, and analyze the graphs important features. Interpret each feature in terms of PlayStation 4s. [HINT: Recall that the average cost function is C(x)=c(x)/x.]41ESubprime Mortgage Debt during the Housing Bubble During the real estate run-up in 20002008 the value of sub- prime (normally classified as risky) mortgage debt outstanding in the United States was approximately A(t)=1,3501+4.2(1.7)tbilliondollars(0t8) t years after the start of 2000. Graph the derivative A(t)ofA(t) usingan extended domain of 0t15. Determine the approximate coordinates of the maximum, and determine the behavior of A(t) atinfinity. What do the answers tell you?43E44E45E46E47E48E49E50ERewrite the statements and questions in Exercises 18 in mathematical notation. [HINT: See Quick Examples 1 and 2.] The population P is currently 10,000 and growing at a rate of 1,000 per year.Rewrite the statements and questions in Exercises 18 in mathematical notation. [HINT: See Quick Examples 1 and 2.] There are currently 400 cases of Bangkok flu, and the number is growing by 30 new cases every month.Rewrite the statements and questions in Exercises 18 in mathematical notation. [HINT: See Quick Examples 1 and 2.] The annual revenue of your tie-dyed T-shirt operation is currently $7,000 but is decreasing by $700 each year. How fast are annual sales changing?Rewrite the statements and questions in Exercises 18 in mathematical notation. [HINT: See Quick Examples 1 and 2.] A ladder is sliding down a wall so that the distance between the top of the ladder and the floor is decreasing at a rate of 3 ft/sec. How fast is the base of the ladder receding from the wall?5E6E7E8ESunspots The area of a circular sunspot is growing at a rate of 1,200km2/sec. a. How fast is the radius growing at the instant when it equals 10,000 kilometers? [HINT: See Example 1.] b. How fast is the radius growing at the instant when the sunspot has an area of 640,000 square kilometers? [HINT: Use the area formula to determine the radius at that instant.]Puddles The radius of a circular puddle is growing at a rate of 5 cm/sec. a. How fast is its area growing at the instant when the radius is 10 centimeters? [HINT: See Example 1.] b. How fast is the area growing at the instant when it equals 36 square centimeters? [HINT: Use the area formula to determine the radius at that instant.]11E12EEnd of the Earth In 5 billion years the Sun will have run out of hydrogen fuel and begin to expand into a red giant, eventually engulfing the Earth and causing it to spiral into the core of the Sun 7.5 billion years from now. At that point, the Suns radius will be around 93 million miles and increasing at a rate of around 0.003 mph. How fast will its volume be increasing? (Round your answer to three significant digits.) [HINT: See the hint for Exercise 11.]14ESliding Ladders The base of a 50-foot ladder is being pulled away from a wall at a rate of 10 ft/sec. How fast is the top of the ladder sliding down the wall at the instant when the base of the ladder is 30 feet from the wall? [HINT: See Example 2.]16E17E18E19EAverage Cost Repeat Exercise 19, using the revised average cost function C(x)=150,000x1+20+0.01xdollarsperplayer, [HINT: See Example 3.]21ESupply The number of retro portable CD players you are prepared to supply to a retail outlet every week is given by the formula q=0.1p2+3p where p is the price it offers you. The retail outlet is currently offering you $40 per CD player. If the price it offers decreases at a rate of $2 per week, how will this affect the number you supply?Revenue You can now sell 50 cups of lemonade per week at 30 per cup, but demand is dropping at a rate of 5 cups per week each week. Assuming that raising the price does not affect demand, how fast do you have to raise your price if you want to keep your weekly revenue constant? [HINT: Revenue=PriceQuantity.]24E25EOil Expenditures Daily oil imports to the United States from Mexico can be approximated by q(t)=39t2+800t-3,000thousandbarrels(8t13) where t is time in years since the start of 2000. At the start of 2012 the price of oil was $105 per barrel and increasing at a rate of $70 per year. How fast was (daily) oil expenditure for imports from Mexico changing at that time?Resource Allocation Your company manufactures automobile alternators, and production is partially automated through the use of robots. To meet production deadlines, your company calculates that the numbers of laborers and robots must satisfy the constraint xy=10,000, where x is the number of laborers and y is the number of robots. Your company currently uses 400 robots and is increasing robot deployment at a rate of 16 per month. How fast is it laying off laborers? [HINT: See Example 4.]28E29E30E31EDemand The demand equation for rubies at Royal Ruby Retailers is q+43p=80, where q is the number of rubies RRR can sell per week at p dollars per ruby. RRR finds that the demand for its rubies is currently 20 rubies per week and is dropping at a rate of one ruby per week. How fast is the price changing?33E34EBaseball A baseball diamond is a square with side 90 feet A batter at home base hits the ball and runs toward first base at a speed of 24 ft/sec. At what rate is his distance from third base increasing when he is halfway to first base?36EMovement along a Graph A point on the graph of y=1/x is moving along the curve in such a way that its x-coordinate is increasing at a rate of 4 units per second. What is happening to the y-coordinate at the instant the y-coordinate is equal to 2?38E39E40E41E42ECones A right circular conical vessel is being filled with green industrial waste at a rate of 100m3/sec. How fast is the level rising after 200 cubicmeters have been poured in? The cone has a height of 50 meters and a radius of 30 meters at its brim. (The volume of a cone of height h and cross-sectional radius r at its brim is given by V=13r2h.)More Cones A circular conical vessel is being filled with ink at a rate of 10cm3/sec. How fast is the level rising after 20 cubic centimeters have been poured in? The cone has height 50 centimeters and radius 20 centimeters at its brim. (The volume of a cone of height h and cross-sectional radius r at its brim is given by V=13r2h.)45E46E47EComputers vs. Income Refer back to the model in Exercise 47. It is 1995, and the average number of computers per household in your town is 0.5 and is increasing at a rate of 0.02 computers per household per year. What is the average household income in your town, and how fast is it increasing? (Round your answers to the nearest $10).49E50E51E52E53E54E55E56E57E58E59E60E61E62EDemand for Oranges The weekly sales of Honolulu Red Oranges is given by q=1,00020p. Calculate the price elasticity of demand when the price is $30 per orange (yes, $30 per orange). Interpret your answer. Also, calculate the price that gives a maximum weekly revenue, and find this maximum revenue. [HINT: See Example 1.]Demand for Oranges Repeat Exercise 1 for weekly sales of 1,00010p. [HINT: See Example 1.]Demand for Smartphones Worldwide annual sales of smart- phones in 20122013 were approximately q=6p+3,030 millionphones at a selling price of $p per phone. a. Obtain a formula for the price elasticity of demand E. b. In 2013 the actual selling price was $335 per phone. What was the corresponding price elasticity of demand? Interpret your answer. c. Use your formula for E to determine the selling price that would have resulted in the largest annual revenue. What, to the nearest $10 million, would have been the resulting annual revenue?4E5EMonorail Services The demand for monorail service in Las Vegas in 2005 could be approximated by q=4,500p+41,500 ridesper day when the fare was $p. a. In September 2005 the Las Vegas monorail increased the price from $3 per ride to $5 per ride. What was the effect on the price elasticity of demand? What does the answer suggest about the fare corresponding to maximum daily revenue? b. What would you have advised the Las Vegas Monorail Company to charge to maximize total daily revenue, and what would the resulting daily ridership and revenue have been?Tissues The consumer demand equation for tissues is given by q=(100p)2, where p is the price per case of tissues and q is the demand in weekly sales. a. Determine the price elasticity of demand E when the price is set at $30, and interpret your answer. b. At what price should tissues be sold to maximize the revenue? c. Approximately how many cases of tissues would be demanded at that price?Bodybuilding The consumer demand curve for Professor Stefan Schwarzenegger dumbbells is given by q=(1002p)2, where p is the price per dumbbell and q is the demand in weekly sales. Find the price Professor Schwarzenegger should charge for his dumbbells to maximize revenue.T-Shirts The Physics Club sells E=mc2 T-shirts at the local flea market. Unfortunately, the clubs previous administration has been losing money for years, so you decide to do an analysis of the sales. A quadratic regression based on old sales data reveals the following demand equation for the T-shirts: q=2p2+33p(9p15). Here, p is the price the club charges per T-shirt and q is the number it can sell each day at the flea market. a. Obtain a formula for the price elasticity of demand for E=mc2 T-shirts. b. Compute the elasticity of demand if the price is set at $10 per shirt. Interpret the result. c. How much should the Physics Club charge for the T-shirts to obtain the maximum daily revenue? What will this revenue be?Comics The demand curve for original Iguanawoman comics is given by q=(400p)2100(0p400), where q is the number of copies the publisher can sell per week if it sets the price at $p. a. Find the price elasticity of demand when the price is set at $40 per copy. b. Find the price at which the publisher should sell the comics to maximize weekly revenue. c. What, to the nearest $1, is the maximum weekly revenue the publisher can realize from sales of Iguana- woman comics?E-Readers The demand for Amazons Kindle e-reader can be approximated by q(p)=21e0.01pmillionunitsperyear(50p400), where p is the price charged by Amazon. Obtain a formula for price elasticity of demand E, and calculate its value at the two endpoints of the given range of prices. Is the price that would maximize annual revenue within the range of prices shown? How would you know this without calculating that price?Monorail Service on Mars The demand for monorail service on the Utarek monorail, which links the three urbynes (or districts) of Utarek on Mars, can be approximated by q(p)=21e0.7pmillionridesperday(3p5), where p is the cost per ride in zonars (Z). Obtain a formula for price elasticity of demand E, and calculate its value at the two endpoints of the given range of prices. Is the price that would maximize daily revenue within the range of prices shown? How would you know this without calculating that price?Corn In the 1930s the economist Henry Schultz devised the following demand function for corn: p=6,570,000q1.3, where q is the number of bushels of corn that could be sold at p dollars per bushel in one year. Express q as a function of p, and find the price elasticity of demand if the price was set at $1.50 per bushel. Interpret the result.Demand for Fried Chicken A fried chicken franchise finds that the demand equation for its new roast chicken product, Roasted Rooster, is given by p=40q1.5, where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price. Express q as a function of p, and find the price elasticity of demand when the price is set at $4 per serving. Interpret the result.Paint-By-Number The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula q=100e3p2+p, where q is the demand in monthly sales and p is the retail price in hundreds of yen. a. Determine the price elasticity of demand E when the retail price is set at 300, and interpret your answer. b. At what price will revenue be a maximum? c. Approximately how many paint-by-number sets will be sold per month at the price in part (b)?16E17E18EHyperbolic Demand Functions A general hyperbolic demand function has the form x q=kpr (r and k nonzero constants). a. Obtain a formula for the price elasticity of demand at unit price p. b. How does E vary with p? c. What does the answer to part (b) say about the model?Quadratic Demand Functions A general quadratic demand function has the form q=ap2+bp+c (a, b, and c constants with a0 ). a. Obtain a formula for the price elasticity of demand at a unit price p. b. Obtain a formula for the price or prices that could maximize revenue.Modeling Linear Demand You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize its weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form q=mp+b,, where p is the price per hamburger, q is the demand in weekly sales, and m and b are certain constants you must determine. a. Your market studies reveal the following sales figures: When the price is set at $2.00 per hamburger, the sales amount to 3,000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero. Use these data to calculate the demand equation. b. Now estimate the unit price that maximizes weekly revenue, and predict what the weekly revenue will be at that price.Modeling Linear Demand You have been hired as a marketing consultant by Big Book Publishing, Inc., and you have been approached to determine the best-selling price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form q=mp+b, where p is the price per book, q is the demand in annual sales, and m and b are certain constants you must determine. a. Your market studies reveal the following sales figures: When the price is set at $50.00 per book, the sales amount to 10,000 per year; when the price is set at $80.00 per book, the sales drop to 1,000 per year. Use these data to calculate the demand equation. b. Now estimate the unit price that maximizes annual revenue and predict what Big Book Publishing, Inc.s annual revenue will be at that price.Modeling Exponential Demand As the new owner of a supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to set the price so that your revenue from selling it is as large as possible. Previous sales figures of the cheese are shown in the following table: PriceperPound,p $3.00 $4.00 $5.00 MonthlySales,q(Pounds) 407 287 223 a. Use the sales figures for the prices $3 and $5 per pound to construct a demand function of the form q=Aebp, where A and b are constants you must determine. (Round A and b to two significant digits.) b. Use your demand function to find the price elasticity of demand at each of the prices listed. c. At what price should you sell the cheese to maximize monthly revenue? d. If your total inventory of cheese amounts to only 200 pounds and it will spoil 1 month from now, how should you price it to receive the greatest revenue? Is this the same answer you got in part (c)? If not, give a brief explanation.24EIncome Elasticity of Demand: Live Drama The likelihood that a child will attend a live theatrical performance can be modeled by q=0.01(0.0078x2+1.5x+4.1)(15x100) Here, q is the fraction of children with annual household income x thousand dollars who will attend a live dramatic performance at a theater during the year. Compute the income elasticity of demand at an income level of $20,000 and interpret the result. (Round your answer to two significant digits.) [HINT: See Example 2.]26EIncome Elasticity of Demand: Broadband in 2010 The following graph shows the percentage q of people in households with annual income x thousand dollars using broadband Internet access in 2010, together with the exponential curve q q=74e0.021x+92 a. Find an equation for the income elasticity of demand for broadband usage, and use it to compute the elasticity for a household with annual income $100,000 to two decimal places. Interpret the result. b. What does the model predict as the elasticity of demand for households with very large incomes?28EIncome Elasticity of Demand: Computer Usage in the 1990s The following graph shows the probability q that a household in the 1990s with annual income x dollars had a computer, together with the logarithmic curve q=0.3454lnx3.047. a. Compute the income elasticity of demand for computers, to two decimal places, for a household income of $60,000, and interpret the result. b. As household income increases, how is income elasticity of demand affected? c. How reliable is the given model of demand for incomes well above $120,000? Explain. d. What can you say about E for incomes much larger than those shown?30E31EPrice Elasticity of Supply Given a supply equation of the form q=f(p), the associated price elasticity of supply is defined as the percentage rate of increase of supply per percentage increase in price: E=dqdppq. (Note that the formula is the same as for price elasticity of demand except for the sign.) Exercises 31 and 32 are based on this formula Saudi Crude Oil Supply: Low Prices For crude oil prices point when of at most $20 per barrel the supply by Saudi Arabia can be approximated by q=0.34p+1.2millionbarrelsperday(12p20), where p is the price per barrel. Calculate the price elasticity of supply when the price of oil is $15 per barrel. What does the answer tell you about Saudi oil production?Income Elasticity of Demand (based on a question on the GRE Economics Test) If Q=aPY is the individuals demand function for a commodity, where P is the (fixed) price of the commodity, Y is the individuals income, and a, , and are parameters, explain why canbe interpreted as the income elasticity of demand.34E35E36E37E38E39E40EIn Exercises 118, evaluate the indefinite integral. (x210x+2)dx2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE