   Chapter 14.4, Problem 31E 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: E a t − i n : x = 6000 − 1000 p x T a k e o u t : y = 10 , 000 − 2000 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.(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 $3.25 each. They sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+1.2x where 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=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. They sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+1.2x where 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=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). Here x is the eat in and take out chicken dinners.

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+1.2x where x is the total number of eat-in and take-out chicken dinners.

Substitute 6x1000 for px, 5y2000 for py, and 500+1

(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. They sells 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+1.2x where 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=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 Whether it would 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. Sea Islands offers both Eat-in and Take-out chicken dinners at \$3.25 each. They sell 6250 dinners per week and has short-run weekly costs (in dollars) given by C=500+1.2x where 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=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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