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Remember: number problems clearly, show all of your work, and circle your final answer.

(11)  Last month the Financial Times reported that Japanese firm Akagi Nyuguyo was increasing the price of its popular children’s ice cream bar Garigari-kun for the first time in 25 years, from ¥60 to ¥70. Suppose that a 7-11 store typically sold 500 ice cream bars a day, but now expects to sell only 400 bars a day. Assuming a linear relationship between price (p) and quantity demanded (x), write down the linear price-demand function p(x) (hint: y−y1 = m(x−x1)).

(12)  Based on your insight from question (11), Akagi Nyuguyo has hired you as their analyst. They need to know the following information:

(a) What sales per 7-11 store will maximize revenue, R(x)?

(b) What is the maximum revenue per store?

(13)  Pleased with your work, the managers at Akagi Nyuguyo ask for a more complete analysis of their business. Using linear regression models, you estimate a total cost function C(x) per 7-11 retailer with a fixed cost of ¥5000 (for freezers, electricity, and marketing) and variable costs of ¥5 (for sticks, packaging, etc.) per bar.

(a) Derive the ice cream bar profit function P(x) per store.

(b) Graph P(x).

(14)  Lastly, your Japanese employers need to know the following:

(a) What sales per 7-11 store will maximize profit, P(x)?

(b) What is the marginal average profit function, per store? (hint: P ′(x) = d P (x)) dx 

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