Chapter 14.4, Problem 32E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

The manager of the Sea Islands Chicken Shack is interested in finding new ways to improve sales and profitability. Currently, Sea Islands offers both “eat-in” and “take-out" chicken dinners at $3.25 each, sells 6250 dinners per week, and has short-run weekly costs (in dollars) given by

where

The Sea Islands manager recently commissioned a local consulting firm to study the eat-in and take-out market demand per week. The study results provided the following weekly demand estimates for each of the two market segments:

**Eat-in:**
**Take out:**

where
*y* is the number of take-out dinners, with

When eat-in and take-out dinners are considered separately, Sea Islands Chicken Shack’s short-run weekly cost function becomes

(a) Use this revised cost function to find the number of eat-in and take-out chicken dinners that would give maximum profit.

(b) What price should be charged for each type of dinner now, and what is the maximum weekly profit?

(c) Based on the results of parts (a) and (b), which is the best pricing strategy? Explain.

**(a)**

To determine

**To calculate:** The number of dinners expected to sell per week to each type of customers to maximize weekly profit if Sea Islands Chicken Shack prices chicken dinners differently for eat-in and take-out customers. Sea Islands offers both eat-in and take-out chicken dinners at *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

Explanation

**Given Information:**

Sea Islands offers both eat-in and take-out chicken dinners at *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

**Formula used:**

To calculate relative maxima and minima of the

(1) Find the partial derivatives

(2) Find the critical points, that is, the point(s) that satisfy

(3) Then find all the second partial derivatives and evaluate the value of *D* at each critical point, where

(a) If

(b) If

For a function *f* with respect to *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant and the partial derivative of *f* with respect to *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant. The partial derivative of *f* with respect to *x* is denoted by *y* is denoted by

For a function

(1) When both derivatives are taken with respect to *x* is

(2) When both derivatives are taken with respect to *y* is

(3) When first derivative is taken with respect to *x* and second derivative is taken with respect to *y* is

(4) When first derivative is taken with respect to *y* and second derivative is taken with respect to *x* is

Power of

Chain rule for function

Constant function rule for a constant

Coefficient rule for a constant

**Calculation:**

The profit function is given by

The demand for the market segment Eat-in is

Rewrite

The demand for the market segment Take-out is

Rewrite

Also, the weekly costs (in dollars) is given by

Substitute

**(b)**

To determine

**To calculate:** The prices Sea Islands should charge each market segment to maximize the total weekly profit and the maximum profit. Sea Islands offers both eat-in and take-out chicken dinners at *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

**(c)**

To determine

The best pricing strategy based on the results of part (a) and part (b). Sea Islands offers both eat-in and take-out chicken dinners at *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

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