The total cost of production, in thousands of dollars, is C(q) = q3 12q2 + 60q, where q

For each cost function in Figure 4.62, is there a value of q at which average cost is minimized? If so, approximately where? Explain your answer. q (quantity) (cost) C (a) q (quantity) (cost) C (b) Figure 4.62

Figure 4.63 shows cost with q = 10,000 marked. (a) Find the average cost when the production level is 10,000 units and interpret it. (b) Represent your answer to part (a) graphically. (c) At approximately what production level is average cost minimized? 10,000 10,000 20,000 q C(q) $ Figure 4.63

The graph of a cost function is given in Figure 4.64. (a) At q = 25, estimate the following quantities and represent your answers graphically. (i) Average cost (ii) Marginal cost (b) At approximately what value of q is average cost minimized? 10 20 30 40 50 100 200 300 400 500 C(q) q $ Figure 4.64

The cost of producing q items is C(q) = 2500+12q dollars. (a) What is the marginal cost of producing the 100th item? the 1000th item? (b) What is the average cost of producing 100 items? 1000 items?

The cost function is C(q) = 1000+20q. Find the marginal cost to produce the 200th unit and the average cost of producing 200 units.

Graph the average cost function corresponding to the total cost function shown in Figure 4.65. 500 1000 q (quantity) cost C Figure 4.65

The total cost of production, in thousands of dollars, is C(q) = q3 12q2 + 60q, where q is in thousands and 0 q 8. (a) Graph C(q). Estimate visually the quantity atwhich average cost is minimized. (b) Determine analytically the exact value of q at which average cost is minimized.

You are the manager of a firm that produces slippers that sell for $20 a pair. You are producing 1200 pairs of slippers each month, at an average cost of $2 each. The marginal cost at a production level of 1200 is $3 per pair. (a) Are you making or losing money? (b) Will increasing production increase or decrease your average cost? Your profit? (c) Would you recommend that production be increased or decreased?

The average cost per item to produce q items is given by a(q) = 0.01q2 0.6q + 13, for q > 0. (a) What is the total cost, C(q), of producing q goods? (b) What is the minimum marginal cost? What is the practical interpretation of this result? (c) At what production level is the average cost a minimum? What is the lowest average cost? (d) Compute the marginal cost at q = 30. How does this relate to your answer to part (c)? Explain this relationship both analytically and in words.

The marginal cost at a production level of 2000 units of an item is $10 per unit and the average cost of producing 2000 units is $15 per unit. If the production level were increased slightly above 2000, would the following quantities increase or decrease, or is it impossible to tell? (a) Averagecost (b) Profit

An agricultural worker in Uganda is planting clover to increase the number of bees making their home in the region. There are 100 bees in the region naturally, and for every acre put under clover, 20 more bees are found in the region. (a) Draw a graph of the total number, N(x), of bees as a function of x, the number of acres devoted to clover. (b) Explain, both geometrically and algebraically, the shape of the graph of: (i) The marginal rate of increase of the number of bees with acres of clover, N(x). (ii) The average number of bees per acre of clover, N(x)/x.

A developer has recently purchased a laundromat and an adjacent factory. For years, the laundromat has taken pains to keep the smoke from the factory from soiling the air used by its clothes dryers. Now that the developer owns both the laundromat and the factory, she could install filters in the factorys smokestacks to reduce the emission of smoke, instead of merely protecting the laundromat from it. The cost of filters for the factory and the cost of protecting the laundromat against smoke depend on the number of filters used, as shown in the table. Number of filters Total cost of filters Total cost of protecting laundromat from smoke 0 $0 $127 1 $5 $63 2 $11 $31 3 $18 $15 4 $26 $6 5 $35 $3 6 $45 $0 7 $56 $0 (a) Make a table which shows, for each possible number of filters (0 through 7), the marginal cost of the filter, the average cost of the filters, and the marginal savings in protecting the laundromat from smoke. (b) Since the developer wishes to minimize the total costs to both her businesses, what should she do? Use the table from part (a) to explain your answer. (c) What should the developer do if, in addition to the cost of the filters, the filters must be mounted on a rack which costs $100? (d) What should the developer do if the rack costs $50?

Show analytically that if marginal cost is less than average cost, then the derivative of average cost with respect to quantity satisfies a(q) < 0.

Show analytically that if marginal cost is greater than average cost, then the derivative of average cost with respect to quantity satisfies a(q) > 0.

A reasonably realisticmodel of a firms costs is given by the short-run Cobb-Douglas cost curve C(q) = Kq1/a + F, where a is a positive constant, F is the fixed cost, and K measures the technology available to the firm. (a) Show that C is concave down if a > 1. (b) Assuming that a < 1, find what value of q minimizes the average cost.