   Chapter 14.4, Problem 32E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

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 C = 500 + 1.2 x 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:Eat-in: x = 6 0 0 0 — 1 0 0 0 p x Take out: y = 1 0 , 0 0 0 − 2 0 0 0 p y where x is the number of eat-in dinners, with p x as the price of each, and y is the number of take-out dinners, with p y as the price of each. Use this information in Problems 31 and 32.When eat-in and take-out dinners are considered separately, Sea Islands Chicken Shack’s short-run weekly cost function becomes C = 500 + 0.60 x + 1.60 y (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$3.25 each, sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+0.60x+1.60y. The demand estimates for each of the two market segments: Eat-in: x=60001000px, Take-out: y=10,0002000py where x is the number of eat-in dinners, with px as the price of each, and y is the number of take-out dinners, with py as the price of each.

Explanation

Given Information:

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 C=500+0.60x+1.60y. The demand estimates for each of the two market segments: Eat-in: x=60001000px, Take-out: y=10,0002000py where x is the number of eat-in dinners, with px as the price of each, and y is the number of take-out dinners, with py as the price of each. Formula used: To calculate relative maxima and minima of the z=f(x,y), (1) Find the partial derivatives zx and zy. (2) Find the critical points, that is, the point(s) that satisfy zx=0 and zy=0. (3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)(zxy)2=2zx22zy2(2zxy)2. (a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0. (b) If D<0, then neither a relative maximum nor a relative minimum occurs. For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to 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 f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by fx and with respect to y is denoted by fy. For a function z(x,y), the second partial derivative, (1) When both derivatives are taken with respect to x is zxx=2zx2=x(zx). (2) When both derivatives are taken with respect to y is zyy=2zy2=y(zy). (3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx). (4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy). Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1. Chain rule for function f(x)=u(v(x)) is f(x)=u(v(x))v(x). Constant function rule for a constant c is such that, if f(x)=c then f(x)=0. Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x). Calculation: The profit function is given by P(x,y)=pxx+pyyC(x,y). The demand for the market segment Eat-in is x=60001000px. Rewrite x=60001000px in terms of px. So, px=6000x1000=6x1000 The demand for the market segment Take-out is y=10,0002000py. Rewrite y=10,0002000py in terms of py. So, py=10,000y2000=5y2000 Also, the weekly costs (in dollars) is given by C=500+0.60x+1.60y. Substitute 6x1000 for px, 5y2000 for py, and 500+0.60x+1.60y for C(x,y) in P(x,y)=pxx+pyyC(x,y) (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$3.25 each, sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+0.60x+1.60y. The demand estimates for each of the two market segments: Eat-in: x=60001000px, Take-out: y=10,0002000py where x is the number of eat-in dinners, with px as the price of each, and y is the number of take-out dinners, with py as the price of each.

(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 \$3.25 each, sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+0.60x+1.60y. The demand estimates for each of the two market segments: Eat-in: x=60001000px, Take-out: y=10,0002000py where x is the number of eat-in dinners, with px as the price of each, and y is the number of take-out dinners, with py as the price of each.

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