The demand equation for a product is p = 45 0.01q. Write the revenue as a function of q

Figure 4.49 shows cost and revenue. For what production levels is the profit function positive? Negative? Estimate the production at which profit is maximized. 5 10 15 100 200 300 400 C(q) R(q) q (thousands) $ (thousands) Figure 4.49

The revenue from selling q items is R(q) = 500q q2, and the total cost is C(q) = 150+10q. Write a function that gives the total profit earned, and find the quantity which maximizes the profit.

Revenue is given by R(q) = 450q and cost is given by C(q) = 10,000 + 3q2. At what quantity is profit maximized? What is the total profit at this production level?

Using the cost and revenue graphs in Figure 4.50, sketch the following functions. Label the points q1 and q2. (a) Totalprofit (b) Marginalcost (c) Marginalrevenue q1 q2 q $ C(q) R(q) Figure 4.50

Figure 4.46 in Section 4.4 shows the points, q1 and q2, where marginal revenue equals marginal cost. (a) On the graph of the corresponding total cost and total revenue functions in Figure 4.51, label the points q1 and q2. Using slopes, explain the significance of these points. (b) Explain in terms of profit why one is a local minimum and one is a local maximum. C(q) R(q) q (quantity) $ Figure 4.51

Let C(q) be the total cost of producing a quantity q of a certain product. See Figure 4.52. (a) What is the meaning of C(0)? (b) Describe in words how the marginal cost changes as the quantity produced increases. (c) Explain the concavity of the graph (in terms of economics). (d) Explain the economic significance (in terms of marginal cost) of the point at which the concavity changes. (e) Do you expect the graph of C(q) to look like this for all types of products? $ q C(q) Figure 4.52

Let C(q) represent the cost, R(q) the revenue, and (q) the total profit, in dollars, of producing q items. (a) If C(50) = 75 and R(50) = 84, approximately how much profit is earned by the 51st item? (b) If C(90) = 71 and R(90) = 68, approximately how much profit is earned by the 91st item? (c) If (q) is a maximum when q = 78, how do you think C(78) and R(78) compare? Explain.

Table 4.2 shows cost, C(q), and revenue, R(q). (a) At approximately what production level, q, is profit maximized? Explain your reasoning. (b) What is the price of the product? (c) What are the fixed costs? Table 4.2 q 0 500 1000 1500 2000 2500 3000 R(q) 0 1500 3000 4500 6000 7500 9000 C(q) 3000 3800 4200 4500 4800 5500 7400

Table 4.3 shows marginal cost, MC, and marginal revenue, MR. (a) Use the marginal cost and marginal revenue at a production of q = 5000 to determine whether production should be increased or decreased from 5000. (b) Estimate the production level that maximizes profit.

Figure 4.53 shows graphs of marginal cost and marginal revenue. Estimate the production levels that could maximize profit. Explain your reasoning. 1000 3000 MC MR q $/unit Figure 4.53

The marginal cost and marginal revenue of a company are MC(q) = 0.03q2 1.4q + 34 and MR(q) = 30, where q is the number of items manufactured. To increase profits, should the company increase or decrease production from each of the following levels? (a) 25 items (b) 50 items (c) 80 items

A manufacturing process has marginal costs given in the table; the item sells for $30 per unit. At how many quantities, q, does the profit appear to be amaximum? Inwhat intervals do these quantities appear to lie? q 0 10 20 30 40 50 60 MC ($/unit) 34 23 18 19 26 39 58

Cost and revenue functions are given in Figure 4.54. Approximately what quantity maximizes profits? 0 7000 20000 40000 60000 C R q $ Figure 4.54

Cost and revenue functions are given in Figure 4.54. (a) At a production level of q = 3000, is marginal cost or marginal revenue greater? Explain what this tells you about whether production should be increased or decreased. (b) Answer the same questions for q = 5000.

When production is 2000, marginal revenue is $4 per unit and marginal cost is $3.25 per unit. Do you expect maximum profit to occur at a production level above or below 2000? Explain.

Revenue and cost functions for a company are given in Figure 4.55. (a) Estimate the marginal cost at q = 400. (b) Should the company produce the 500th item? Why? (c) Estimate the quantity which maximizes profit. 200 400 600 0 200 400 600 R(q) C(q) q $ Figure 4.55

A company estimates that the total revenue, R, in dollars, received from the sale of q items is R = ln(1+1000q2). Calculate and interpret the marginal revenue if q = 10.

The demand equation for a product is p = 45 0.01q. Write the revenue as a function of q and find the quantity that maximizes revenue. What price corresponds to this quantity? What is the total revenue at this price?

The demand for tickets to an amusement park is given by p = 700.02q, where p is the price of a ticket in dollars and q is the number of people attending at that price. (a) What price generates an attendance of 3000 people? What is the total revenue at that price? What is the total revenue if the price is $20? (b) Write the revenue function as a function of attendance, q, at the amusement park. (c) What attendance maximizes revenue? (d) What price should be charged to maximize revenue? (e) What is the maximum revenue? Can we determine the corresponding profit?

An ice cream company finds that at a price of $4.00, demand is 4000 units. For every $0.25 decrease in price, demand increases by 200 units. Find the price and quantity sold that maximize revenue.

At a price of $8 per ticket, a musical theater group can fill every seat in the theater, which has a capacity of 1500. For every additional dollar charged, the number of people buying tickets decreases by 75. What ticket price maximizes revenue?

The demand equation for a quantity q of a product at price p, in dollars, is p = 5q + 4000. Companies producing the product report the cost, C, in dollars, to produce a quantity q is C = 6q +5 dollars. (a) Express a companys profit, in dollars, as a function of q. (b) What production level earns the company the largest profit? (c) What is the largest profit possible?

(a) Production of an item has fixed costs of $10,000 and variable costs of $2 per item. Express the cost, C, of producing q items. (b) The relationship between price, p, and quantity, q, demanded is linear. Market research shows that 10,100 items are sold when the price is $5 and 12,872 items are sold when the price is $4.50. Express q as a function of price p. (c) Express the profit earned as a function of q. (d) How many items should the company produce to maximize profit? (Give your answer to the nearest integer.) What is the profit at that production level?

An online seller of knitted sweaters finds that it costs $35 to make her first sweater. Her cost for each additional sweater goes down until it reaches $25 for her 100th sweater, and after that it starts to rise again. If she can sell each sweater for $35, is the quantity sold that maximizes her profit less than 100? Greater than 100?

A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing $45 per foot along three sides and fencing costing $20 per foot along the fourth side. Find the minimum total cost.

You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by $0.25 per chair (on the whole order) for every additional chair over 300 ordered. What are the largest and smallest revenues your company can make under this deal?

A warehouse selling cement has to decide how often and in what quantities to reorder. It is cheaper, on average, to place large orders, because this reduces the ordering cost per unit. On the other hand, larger orders mean higher storage costs. The warehouse always reorders cement in the same quantity, q. The total weekly cost, C, of ordering and storage is given by C = a q + bq, where a, b are positive constants. (a) Which of the terms, a/q and bq, represents the ordering cost and which represents the storage cost? (b) What value of q gives the minimum total cost?

A demand function is p = 4002q, where q is the quantity of the good sold for price $p. (a) Find an expression for the total revenue, R, in terms of q. (b) Differentiate R with respect to q to find the marginal revenue, MR, in terms of q. Calculate the marginal revenue when q = 10. (c) Calculate the change in total revenue when production increases from q = 10 to q = 11 units. Confirm that a one-unit increase in q gives a reasonable approximation to the exact value of MR obtained in part (b).

A business sells an item at a constant rate of r units per month. It reorders in batches of q units, at a cost of a+bq dollars per order. Storage costs are k dollars per item per month, and, on average, q/2 items are in storage, waiting to be sold. [Assume r, a, b, k are positive constants.] (a) How often does the business reorder? (b) What is the average monthly cost of reordering? (c) What is the total monthly cost, C of ordering and storage? (d) Obtain Wilsons lot size formula, the optimal batch size which minimizes cost.

(a) A cruise line offers a trip for $2000 per passenger. If at least 100 passengers sign up, the price is reduced for all the passengers by $10 for every additional passenger (beyond 100) who goes on the trip. The boat can accommodate 250 passengers. What number of passengers maximizes the cruise lines total revenue? What price does each passenger pay then? (b) The cost to the cruise line for n passengers is 80,000+400n. What is themaximumprofit that the cruise line can make on one trip? How many passengers must sign up for the maximum to be reached and what price will each pay?

A company manufactures only one product. The quantity, q, of this product produced per month depends on the amount of capital, K, invested (i.e., the number of machines the company owns, the size of its building, and so on) and the amount of labor, L, available each month. We assume that q can be expressed as a Cobb-Douglas production function: q = cKL, where c, , are positive constants, with 0 < < 1 and 0 < < 1. In this problemwewill see how the Russian government could use a Cobb-Douglas function to estimate how many people a newly privatized industry might employ. A company in such an industry has only a small amount of capital available to it and needs to use all of it, so K is fixed. Suppose L is measured in man-hours per month, and that each man-hour costs the company w rubles (a ruble is the unit of Russian currency). Suppose the company has no other costs besides labor, and that each unit of the good can be sold for a fixed price of p rubles. How many man-hours of labor per month should the company use in order to maximize its profit?

A company can produce and sell f(L) tons of a product per month using L hours of labor per month. The wage of the workers is w dollars per hour, and the finished product sells for p dollars per ton. (a) The function f(L) is the companys production function. Give the units of f(L). What is the practical significance of f(1000) = 400? (b) The derivative f(L) is the companys marginal product of labor. Give the units of f(L). What is the practical significance of f(1000) = 2? (c) The real wage of the workers is the quantity of product that can be bought with one hours wages. Show that the real wage is w/p tons per hour. (d) Show that the monthly profit of the company is (L) = pf(L) wL. (e) Show that when operating at maximum profit, the companys marginal product of labor equals the real wage: