Concept explainers
Business The annual cost C (in thousands of dollars)and revenue R (in thousands of dollars) for a company each year from 2010 through 2016 can be approximated by the models
where t is the year, with
(a) Write a
(b) Use a graphing utility to graph C, R, and P in the same viewing window.
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Chapter P Solutions
Trigonometry (MindTap Course List)
- Profit The yearly profit P for a widget producer is a function of the number n of widgets sold. The formula is P=180+100n4n2. Here P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to level of 20 thousand widgets sold. a. Make a graph of P versus n. b. Calculate P0 and explain in practical terms. What your answer means. c. What profit will the producer make if 15 thousand widgets are sold?. d. The break-even point is the sales level at which the profit is 0. Approximate the break-even point for this widget producer. e. What is the largest profit possible?arrow_forwardMaximum Revenue A small theater has a seating capacity of 2000.When the ticket price is 20,attendance is 1500.For each 1decrease in price, attendance increases by 100. (a) Write the revenue R of the theater as a function of ticket price x. (b) What ticket price will yield a maximum revenue? What is the maximum revenue?arrow_forwardProfit A company that produces calculators estimates that the profit P(in dollars) from selling a specific model of calculator is given by P=152x3+7545x2169,625,0x45 where xis the advertising expense (in tens of thousands of dollars).For this model of calculator, an advertising expense of $400,000(x=40)results in a profit of $2,174,375. (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company can spend on advertising that results in the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.arrow_forward
- Profit The demand equation for a microwave oven is given by p=1400.0001x, where p is the unit price (in dollars) of the microwave oven and x is the number of units sold. The cost equation for the microwave oven is C=80x+150,000, where C is the total cost (in dollars) and x is the number of units produced. The total profit P obtained by producing and selling x units is modeled by P=xpC. (a) Find the profit function P in terms of x. (b) Find the profit when 250,000 units are sold. (c) Find the unit price when 250,000 units are sold. (d) Find (if possible) the unit price that will yield a profit of 10 million dollars. If not possible, explain why.arrow_forwardPopulation Statistics The table shows the life expectancies of a child (at birth) in the United States for selected years from 1940 through 2010. A model for the life expectancy during this period is y=63.6+0.97t1+0.01t,0r70 Where y represents the life expectancy and t is the time in years, with t=0 corresponding to 1940. (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain (b) Determine the life expectancy in 1990 both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately 70.1. Verify your answer algebraically. (d) Identify the y-intercept of the graph of the model. What does it represent in the context of the problem? (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explainarrow_forwardSeizure of Illegal Drugs The cost C(in millions of dollars) for the federal government to seize p%of an illegal drug as it enters the country is given by C=528p100p,0p100. (a) Use a graphing utility to graph the cost function. (b) Find the costs of seizing 25%,50%,and 75%of the drug. (c) According to the model, is it possible to seize 100%of the drug? Explain.arrow_forward
- Profit The weekly profit P for a widget producer is a function of the number n of widgets sold. The formula is P=2+2.9n0.3n2 Here P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to a level of 7 thousands widgets sold. a.Make a graph of P versus n. b.Calculate P0 and explain in practical terms what your answer means. c.At what sales level is the profit as large as possible?arrow_forwardProfit The profit P, in thousands of dollars that a manufacturer makes is a function of the number N of items produced in a year, and the formula is P=0.2N2+3.6N9. a. Express using functional notation the profit at a production level of 5 items per year, and then calculate that value. b. Determine the two break-even points for this manufacturesthat is, the two production levels at which the profit is zero. c. Determine the maximum profit if the manufacturer can produce at most 20 items in a year.arrow_forwardPopulation Statistics The table shows the life expectancies of a child (at birth) in the United States for selected years from 1940 through 2010. A model for the life expectancy during this period is y=63.6+0.97t1+0.01t,0t70 Where y represents the life expectancy and t is the time in years, with t = 0 corresponding to 1940. (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain. (b) Determine the life expectancy in 1990 both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately 70.1. verify your answer algebraically. (d) Find the y-intercept of the graph of the model. What does it represent in the context of the problem? (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now?arrow_forward
- Functional Response: The amount C of food consumed in a day by a sheep is a function of the amount V of vegetation available, and a model is C=3V50+V, Here C is measured in pounds and V in pounds per acre. This relationship is called the functional response. a. Make a graph of C versus V. Include vegetation levels up to 1000 pounds per acre. b. Calculate C300 and explain in practical terms what your answer means. c. Is this graph concave up or concave down? Explain in practical terms what this means. d. From the graph, it should be apparent that there is a limit to the amount of food consumed as more and more vegetation is available. Find this limiting value of C.arrow_forwardRevenue A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x)=80x0.4x2, where the revenue R(x) is measured in dollars. What is the maximum revenue? and how many units should be manufactured to obtain this maximum?arrow_forwardMaximum Volume An open box of maximum volume is made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) The table shows the volumes V (in cubic centimeters) of the box for various heights x (in centimeters). Use the table to estimate the maximum volume. (b) Plot the points (x,V) from the table in part (a). Does the relation defined by the ordered pairs represent V as a function of x ? (c) Given that V is a function of x, write the function and determine its domain.arrow_forward
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