Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Note Some of the formulas below use the special number
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
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
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. Kleibers law states that for the vast majority of animals, the metabolic rate M is a power function of the weight W, and the power is k=34. a.The Eastern gray squirrel weighs about 1 pound. How does the squirrels metabolic rate compare with that of a 200 -pound man? b.How does a 200- pound mans metabolic rate compare with that of a 130- pound woman? c.Based on your answer to part b, would overeating the same amount for each be more likely to lead to weight gain for the man or for the woman?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Math and the City An article in The New York Times states, "The number of gas stations in a city grows only in proportion to the 0.77 power of population. This means that the approximate number G of gas stations in a city is a power function of the population N, and the power is k=0.77. That is, G=cN0.77, where c is some as yet unknown constant. We measure N in millions. a. If one city is twice as large as another, how do the numbers of gas stations compare? b. The population of Houston, Texas, is 2.2million and, according to Yahoo Local, there are 1239 gas stations in Houston. Use this information to find the value of c. c. Los Angeles has a population of about 3.9million. Using the value of c that you found in part b, estimate the number of gas stations in Los Angeles. Round your answer to the nearest whole number. Note: According to Yahoo Local, the correct number is 2013.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Estimating Wave Height Sailors use the following function to estimate wave height h, in feet, from wind speed w, in miles per hour h=0.02w2 a. Make a graph of wave height versus wind speed. Include wind speeds of up to 25 miles per hour. b. A small boat can sail safely provided wave heights are no more than 4 feet. What range of wind speed will give safe sailing for this boat?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Ship Propellers An ideal diameter d, in feet, of a ships propeller is given by the formula d=ch1/5r3/5. Here h is the horsepower of the engine driving the propeller. r is the maximum number of revolutions per minute of the propeller, and c is a constant. In both parts, give your answer in terms of a percentage. a. If the horsepower is increased by 20 while the number of revolutions per minute remains the same, how is the propeller diameter affected? b. If the horsepower remains the same while the number of revolutions per minute is increased by 20, how is the propeller diameter affected?arrow_forwardReminder 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.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Sales Growth A study of the sales s, in thousands of dollars, of a product as a function of time t, in years, yields the equation of change dsdt=0.3s(4s). This is valid for s less than 5. a.What level of sales will be attained in the long run? b.What is the largest rate of growth in sales?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Length of Skid Marks Versus Speed When a car skids to a stop, the length L, in feet, of the skid marks is related to the speed S, in miles per hour, of the car by the power function L=130hS2. Here the constant h is the friction coefficient, which depends on the road surface. For dry concrete pavement, the value of h is about 0.85. a. If a driver going 55milesperhour on dry concrete jams on the brakes and skids to a stop. how long will the skid marks be? b. A policeman investigating an accident on dry concrete pavement finds skid marks 230feet long. The speed limit in the area is 60milesperhour. Is the driver in danger of getting a speeding ticket? c This part of the problem applies to any road surface, so the value of h is not known. Suppose you are driving at 60milesperhour, but, because of approaching darkness, you wish to slow to a speed that will cut your emergency stopping distance in half. What should your new speed be? Hint: You should use the homogeneity property of power functions here. By what factor should you change your speed to ensure that L changes by a factor of 0.5?arrow_forwardReminderRound 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?arrow_forwardReminder 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?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Travel Time The time T, in hours, required to drive 100miles is a function of the average speed s, in miles per hour. The formula is T=100s. a. Make a graph T versus s covering speeds up to 70milesperhour. b. Calculate T(25) and explain in practical terms what your answer means. c. Explain in practical terms the behavior of the graph near the pole at s=0.arrow_forwardReminderRound 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.arrow_forwardReminder 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.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning