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SECTION 10.3 Optimization in Business and Economics 663 | EXERCISES | 10.3 In Problems 1–4, find the absolute maxima and minima for f (x) on the interval [a, b] f (x) ϭ x Ϫ 2x Ϫ 4x ϩ 2, [Ϫ1, 3] f (x) ϭ x Ϫ 3x ϩ 3, [Ϫ3, 1.5] f (x) ϭ x ϩ x Ϫ x ϩ 1, [Ϫ2, 0] f (x) ϭ x Ϫ x Ϫ x, [Ϫ0.5, 2] MAXIMIZING REVENUE (a) If the total revenue function for a hammer is R ϭ 36x Ϫ 0.01x 2, then sale of how many hammers, x, will maximize the total revenue in dollars? Find the maximum revenue (b) Find the maximum revenue if production is limited to at most 1500 hammers (a) If the total revenue function for a blender is R(x) ϭ 25x Ϫ 0.05x 2, sale of how many units, x, will provide the maximum total revenue in dollars? Find the maximum revenue (b) Find the maximum revenue if production is limited to at most 200 blenders If the total revenue function for a computer is R(x) ϭ 2000x Ϫ 20x Ϫ x 3, find the level of sales, x, that maximizes revenue and find the maximum revenue in dollars A firm has total revenues given by R(x) ϭ 2800x Ϫ 8x Ϫ x dollars 10 11 12 for x units of a product Find the maximum revenue from sales of that product An agency charges $100 per person for a trip to a concert if 70 people travel in a group But for each person above the 70, the charge will be reduced by $1.00 How many people will maximize the total revenue for the agency if the trip is limited to at most 90 people? A company handles an apartment building with 70 units Experience has shown that if the rent for each of the units is $1080 per month, all the units will be filled, but unit will become vacant for each $20 increase in the monthly rate What rent should be charged to maximize the total revenue from the building if the upper limit on the rent is $1300 per month? A cable TV company has 4000 customers paying $110 each month If each $1 reduction in price attracts 50 new customers, find the price that yields maximum revenue Find the maximum revenue If club members charge $5 admission to a classic car show, 1000 people will attend, and for each $1 increase in price, 100 fewer people will attend What price will give the maximum revenue for the show? Find the maximum revenue 13 The function R(x) ϭ R(x) x defines the average revenue for selling x units For R(x) ϭ 2000x ϩ 20x Ϫ x (a) find the maximum average revenue (b) show that R(x) attains its maximum at an x-value where R(x) ϭ MR 14 For the revenue function given by R(x) ϭ 2800x ϩ 8x Ϫ x (a) find the maximum average revenue (b) show that R(x) attains its maximum at an x-value where R(x) ϭ MR MINIMIZING AVERAGE COST 15 If the total cost function for a lamp is C(x) ϭ 250 ϩ 33x ϩ 0.1x dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost 16 If the total cost function for a product is C(x) ϭ 300 ϩ 10x ϩ 0.03x dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost 17 If the total cost function for a product is C(x) ϭ 810 ϩ 0.1x dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost 18 If the total cost function for a product is C(x) ϭ 250 ϩ 6x ϩ 0.1x dollars, producing how many units, x, will minimize the average cost? Find the minimum average cost 19 If the total cost function for a product is C(x) ϭ 100(0.02x ϩ 4)3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize average cost? Find the minimum average cost 20 If the total cost function for a product is C(x) ϭ (x ϩ 5)3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize average cost? Find the minimum average cost 21 For the cost function C(x) ϭ 25 ϩ 13x ϩ x 2, show that average costs are minimized at the x-value where C(x) ϭ MC 22 For the cost function C(x) ϭ 300 ϩ 10x ϩ 0.03x 2, show that average costs are minimized at the x-value where C(x) ϭ MC Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 664 CHAPTER 10 Applications of Derivatives Dollars The graphs in Problems 23 and 24 show total cost functions For each problem: (a) Explain how to use the total cost graph to determine the level of production at which average cost is minimized (b) Determine that level of production C(x) 23 x Units 24 C(x) many items, x, should the firm produce for maximum profit? Find the maximum profit 30 A firm can produce 100 units per week If its total cost function is C ϭ 500 ϩ 1500x dollars and its total revenue function is R ϭ 1600x Ϫ x dollars, how many units, x, should it produce to maximize its profit? Find the maximum profit 31 Marginal revenue and marginal cost The figure shows the graph of a quadratic revenue function and a linear cost function (a) At which of the four x-values shown is the distance between the revenue and the cost greatest? (b) At which of the four x-values shown is the profit largest? (c) At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the cost line? (d) What is the relationship between marginal cost and marginal revenue when profit is at its maximum value? Dollars $ Dollars R(x) C(x) x Units x A MAXIMIZING PROFIT 25 If the profit function for a product is P(x) ϭ 5600x ϩ 85x Ϫ x Ϫ 200,000 dollars, selling how many items, x, will produce a maximum profit? Find the maximum profit 26 If the profit function for a commodity is P ϭ 6400x Ϫ 18x Ϫ 31 x Ϫ 40,000 dollars, selling how many units, x, will result in a maximum profit? Find the maximum profit 27 A manufacturer estimates that its product can be produced at a total cost of C (x) ϭ 45,000 ϩ 100x ϩ x dollars If the manufacturer’s total revenue from the sale of x units is R(x) ϭ 4600x dollars, determine the level of production x that will maximize the profit Find the maximum profit 28 A product can be produced at a total cost C(x) ϭ 800 ϩ 100x ϩ x dollars, where x is the number produced If the total revenue is given by R(x) ϭ 60,000x Ϫ 50x dollars, determine the level of production, x, that will maximize the profit Find the maximum profit 29 A firm can produce only 1000 units per month The monthly total cost is given by C(x) ϭ 300 ϩ 200x dollars, where x is the number produced If the total revenue is given by R(x) ϭ 250x Ϫ 100 x dollars, how B C D Units 32 Marginal revenue and marginal cost The figure shows the graph of revenue function y ϭ R(x) and cost function y ϭ C(x) (a) At which of the four x-values shown is the profit largest? (b) At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve? (c) What is the relationship between marginal cost and marginal revenue when profit is at its maximum value? y = R(x) y = C(x) A B C D x Units Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 10.3 33 A company handles an apartment building with 50 units Experience has shown that if the rent for each of the units is $720 per month, all of the units will be filled, but unit will become vacant for each $20 increase in this monthly rate If the monthly cost of maintaining the apartment building is $12 per rented unit, what rent should be charged per month to maximize the profit? 34 A travel agency will plan a tour for groups of size 25 or larger If the group contains exactly 25 people, the cost is $500 per person However, each person’s cost is reduced by $10 for each additional person above the 25 If the travel agency incurs a cost of $125 per person for the tour, what size group will give the agency the maximum profit? 35 A firm has monthly average costs, in dollars, given by 45,000 Cϭ ϩ 100 ϩ x x where x is the number of units produced per month The firm can sell its product in a competitive market for $1600 per unit If production is limited to 600 units per month, find the number of units that gives maximum profit, and find the maximum profit 36 A small business has weekly average costs, in dollars, of Cϭ 37 38 39 40 100 x ϩ 30 ϩ x 10 where x is the number of units produced each week The competitive market price for this business’s product is $46 per unit If production is limited to 150 units per week, find the level of production that yields maximum profit, and find the maximum profit The weekly demand function for x units of a product sold by only one firm is p ϭ 600 Ϫ 21 x dollars, and the average cost of production and sale is C ϭ 300 ϩ 2x dollars (a) Find the quantity that will maximize profit (b) Find the selling price at this optimal quantity (c) What is the maximum profit? The monthly demand function for x units of a product sold by a monopoly is p ϭ 8000 Ϫ x dollars, and its average cost is C ϭ 4000 ϩ 5x dollars (a) Determine the quantity that will maximize profit (b) Determine the selling price at the optimal quantity (c) Determine the maximum profit The monthly demand function for a product sold by a monopoly is p ϭ 1960 Ϫ 13 x dollars, and the average cost is C ϭ 1000 ϩ 2x ϩ x dollars Production is limited to 1000 units and x is in hundreds of units (a) Find the quantity that will give maximum profit (b) Find the maximum profit The monthly demand function for x units of a product sold by a monopoly is p ϭ 5900 Ϫ 12 x dollars, and its average cost is C ϭ 3020 ϩ 2x dollars If production is limited to 100 units, find the number of units that Optimization in Business and Economics 665 maximizes profit Will the maximum profit result in a profit or loss? 41 An industry with a monopoly on a product has its average weekly costs, in dollars, given by Cϭ 10,000 ϩ 60 Ϫ 0.03x ϩ 0.00001x x The weekly demand for x units of the product is given by p ϭ 120 Ϫ 0.015x dollars Find the price the industry should set and the number of units it should produce to obtain maximum profit Find the maximum profit 42 A large corporation with monopolistic control in the marketplace has its average daily costs, in dollars, given by Cϭ 800 ϩ 100x ϩ x x The daily demand for x units of its product is given by p ϭ 60,000 Ϫ 50x dollars Find the quantity that gives maximum profit, and find the maximum profit What selling price should the corporation set for its product? 43 Coastal Soda Sales has been granted exclusive market rights to the upcoming Beaufort Seafood Festival This means that during the festival Coastal will have a monopoly, and it is anxious to take advantage of this position in its pricing strategy The daily demand function is p ϭ Ϫ 0.0004x and the daily total cost function is C(x) ϭ 800 ϩ 0.2x ϩ 0.0001x where x is the number of units (a) Determine Coastal’s total revenue and profit functions (b) What profit-maximizing price per soda should Coastal charge, how many sodas per day would it expect to sell at this price, and what would be the daily profits? (c) If the festival organizers wanted to set an economically efficient price of $1.25 per soda, how would this change the results from part (b)? Would Coastal be willing to provide sodas for the festival at this regulated price? Why or why not? 44 A retiree from a large Atlanta financial services firm decides to keep busy and supplement her retirement income by opening a small upscale folk art company near Charleston, South Carolina The company, Sand Dollar Art, manufactures and sells in a purely competitive market, and the following monthly market information for x units at $p per unit applies: Demand: Supply: p ϭ 2000 Ϫ 4.5x p ϭ 100 ϩ 0.25x Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... which of the four x-values shown is the distance between the revenue and the cost greatest? (b) At which of the four x-values shown is the profit largest? (c) At which of the four x-values shown is... function y ϭ R(x) and cost function y ϭ C(x) (a) At which of the four x-values shown is the profit largest? (b) At which of the four x-values shown is the slope of the tangent to the revenue curve equal... is the number of units (a) Determine Coastal’s total revenue and profit functions (b) What profit-maximizing price per soda should Coastal charge, how many sodas per day would it expect to sell
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