   Chapter 14.5, Problem 23E 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

Revenue On the basis of past experience, a company has determined that its monthly sales revenue (in dollars) is related to its advertising according to the formula s   =   20 x +   y 2 +   4 x y , where x is the amount spent on print advertising and y is the amount spent on cable advertising. If the company plans to spend $30,000 per month on these two means of advertising, how much should it spend on each method to maximize its monthly sales revenue? To determine To calculate: The cost to be spend on each method to maximize the monthly sales revenue of a company, if the company plans to spend$30,000 per month on the two means of advertising. On the basis of past experience, a company has determined that its monthly sales revenue (in dollars) is related to its advertising according to the formula s=20x+y2+4xy, where x is the amount spent on print advertising and y is the amount spent on cable advertising.

Explanation

Given Information:

A company has determined that its monthly sales revenue (in dollars) is related to its advertising according to the formula s=20x+y2+4xy, where x is the amount spent on print advertising and y is the amount spent on cable advertising. The company plans to spend $30,000 per month on the two means of advertising. Formula used: Lagrange Multipliers Method: According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0, Step 1: Find the critical values of f(x,y) using the new variable λ to form the objective function F(x,y,λ)=f(x,y)+λg(x,y). Step 2: The critical points of f(x,y) are the critical values of F(x,y,λ) which satisfies g(x,y)=0. Step 3: The critical points of F(x,y,λ) are the points that satisfy: Fx=0, Fy=0, and Fλ=0, that is, the points which make all the partial derivatives of zero. For a function f(x,y), the partial derivative of f(x,y) 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(x,y) with respect to y is denoted by fy. Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1. 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: Consider the problem, a company has determined that its monthly sales revenue (in dollars) is related to its advertising according to the formula s=20x+y2+4xy, where x is the amount spent on print advertising and y is the amount spent on cable advertising. The company plans to spend$30,000 per month on the two means of advertising.

If x is the amount spent on print advertising and y is the amount spent on cable advertising, then the total amount spent is x+y. But the company plans to spend \$30,000 per month on the two means of advertising. Thus, x+y=30000.

Thus, the constraint is x+y=30000.

Thus, minimize the function s(x,y)=20x+y2+4xy.

According to the Lagrange multipliers method,

The objective function is F(x,y,λ)=f(x,y)+λg(x,y).

Thus, f(x,y)=20x+y2+4xy and g(x,y)=x+y30,000.

Substitute 20x+y2+4xy for f(x,y) and x+y30,000 for g(x,y) in F(x,y,λ)=f(x,y)+λg(x,y)

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