Chapter 14.4, Problem 31E

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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 *x* is the total number of eat-in and take-out chicken dinners.

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:

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

(a) If Sea Islands Chicken Shack prices chicken dinners differently for eat-in and take-out customers, how many dinners per week would it expect to sell to each type of customer to maximize weekly profit?

(b) What prices should Sea Islands charge each market segment to maximize the total weekly profit, and what is that profit?

(c) Would it be more profitable for Sea Islands to continue charging $3.25 per dinner (and sell 6250 dinners per week) or to change the policy and to price differently for each type of customer? Explain.

**(a)**

To determine

**To calculate:** The number of dinners expected to sell per week to each, Eat-in and take-out, customers to maximize the weekly profit. Sea Islands offers both Eat-in and Take-out chicken dinners at *x* is the total number of eat-in and take-out chicken dinners. The demand estimates for each of the two market segments are: Eat-in: *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 total number of eat-in and take-out chicken dinners. The demand estimates for each of the two market segments are: Eat-in: *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 *x* is the eat in and take out chicken dinners.

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 *x* is the total number of eat-in and take-out chicken dinners.

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 total number of eat-in and take-out chicken dinners. The demand estimates for each of the two market segments are: Eat-in: *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

**(c)**

To determine

Whether it would be more profitable for Sea Islands to continue charging *x* is the total number of eat-in and take-out chicken dinners. The demand estimates for each of the two market segments are: Eat-in: *x* is the number of eat-in dinners, with *y* is the number of take-out dinners, with

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