Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
The Role of the Coefficient in Power Functions with a Negative Power Consider the family of power functions
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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. Composing Functions Use a formula to express y as a function of t if y=3x2+5x and x=t1.arrow_forwardReminder 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_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. 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.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 the answers to two decimal places unless otherwise indicated. A Leaking Can The side of a cylindrical can full of water springs a leak, and the water begins to stream out. See Figure 5.73. The depth H, in inches, of water remaining in the can is a function of the distance D in inches measured from the base of the can at which the stream of water strikes the ground. Here is a table of values of D and H: Distance D, in inches Depth H, in inches 0 1.00 1 1.25 2 2.00 3 3.25 4 5.00 a. Use regression to find a formula for H as a quadratic function of D. b. When the depth is 4 inches, how far from the base of the can will the water stream strike the ground? c. When the water stream strikes the ground 5 inches from the base of the can, what is the depth of water in the can?arrow_forward
- 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.arrow_forwardReminder 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.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning