Can you please show work to help me understand how you got the answer for both the questions posted and the worksheet. Thanks in advance!

 

13. The cost to a manufacturer of building x boats is C(x) = 100,000 + 3,000x+x². Each boat sells for $5,000. (a) Find the revenue and profit as functions of the number of boats that the manufacturer builds. (b) Calculate the marginal profit as a function of the number of boats built. (c) Suppose 1000 boats are built. Calculate the profit. (d) When 1000 boats are built, is the profit increasing (with more production)? decreasing?

 

14. The cost to a store of purchasing x cereal boxes is C(x)=1,000 +x+0.001x². Each cereal box sells for $4.50 (assume all the purchased boxes do get sold). (a) Suppose that 1000 boxes of cereal are sold in one year. Calculate the profit from these sales. (b)Suppose that 2000 boxes of cereal are sold in one year. Calculate the profit with this new number of sales. (c) Calculate the profit per item with sales as in (a) and as in (b).

 

15. The cost, in dollars, to a store of procuring x bicycles is C(x) = 200,000 +400x+5x². Each bicycle sells for $3,000. Find the revenue as a function of the number of bicycles that the store sells. (a) Find the average cost and average revenue as functions of the number of bicycles sold. (b) Calculate the marginal cost. (c) Suppose 1,300 of these bicycles are sold. Calculate the average cost, average revenue, marginal revenue,and marginal cost. (d) Which of the numbers in part (c) are most important to the store’s decision regarding ordering bicycles? (e) Which of the numbers in part (c) are most important to the store’s balance sheet?

 

16. A kayak school finds that when a certain training class costs $700 there are 100 students per summer who sign up for this class, and that when the class costs $800 only 90 students sign up for a the class. Suppose that the rate at which the number of students taking class declines with the price is constant. Suppose cost of offering this class is $300 per student. (a) Find the revenue, cost, and profit from these classes as functions of the price charged per class. (b) Calculate the marginal profit when the price is $700. (c) Interpret this number in terms of the school’s decision regarding the price of its class.

 

17. A pizza shop finds that when a pizza costs $15, 100 pizzas are sold per day, and when a pizza costs $10, 150 pizzas are sold. Suppose the relation between the price and the number of pizzas sold is linear. (a) Find the revenue as a function of the price charged per pizza. (b) Calculate the marginal revenue when the price is $14. (c) Would the revue be greater with a price of $12 or $14?

 

22. Imagine that we are considering designs for distribution centers. The cost of sending out x items on any given day is C(x)=5,000+2x+0.1x². (These are fixed costs,costs associated directly with the each shipment and the costs associated with logistics for a larger number of parcels.) (a) Calculate the average cost. (b) Calculate the marginal cost. (c) For what number of items is the average cost equal to the marginal cost? (d) What does the previous calculation tell us about the most cost effective size for the distribution centers?

 

23. Imagine that we are considering the best time for replanting an orchard. Annual production from an orchard of age a is X(a)=20a+0.4a²âˆ’0.008a³. (Production is measured in bushels per acre.) (a) Calculate the marginal production rate and the average production rate over the life of the orchard. (b) At what age is the average production rate equal to the marginal production rate? (b) Suppose that we can ignore the cost of replanting. At which age should the orchard be replanted? Explain the reasoning behind this decision.

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