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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Advertising A private golf club is determining how to spend its $8100 advertising budget. The club knows from prior experience that the number of responses A is given by

A =   0.0001 t 2 p r 1.5 where t is the number of cable television ads, p is the number of newspaper ads. and r is the number of radio ads. A cable television ad costs $90, a newspaper ad costs $36, and a radio ad costs $45.

(a) How much should be spent on each type of advertising to obtain the maximum number of responses? (Assume the golf club uses each type of advertising.)

(b) What is the maximum number of responses expected?

(a)

To determine

To calculate: The amount that should be spent on each type of advertising to obtain the maximum number of response.

Explanation

Given Information:

A private golf club is determining how to spend its $8100 advertising budget. The club knows from prior experience that the number of response A=0.0001t2pr1.5. t is the cable television ads cost $90, p is the numer of newspaper ads cost $36, r is the number of radio ads cost $45.

Formula used:

Method of Lagrange multipliers,

If the function f(x,y) contains a maximum or minimum subject to the constraint g(x,y)=0

Then the maximum or minimum can occur at one of the critical numbers of the function F is,

F(x,y,λ)=f(x,y)λg(x,y) where, λ is a Lagrange multiplier.

Steps to determine the minimum or maximum of the function f.

1. Solve the system of equations,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

2. Determine the value of the function f at each solution obtained from the step 1.

The largest value gives the maximum value of function f subject to the constraint g(x,y)=0 and the lowest value gives the minimum value of function f subject to the constraint g(x,y)=0.

Calculation:

Consider the function,

A=0.0001t2pr1.5 Subject to constraint 90t+36p+45r=8100

Maximize A=0.0001t2pr1.5 subject to constraint 90t+36p+45r=8100

F(A)=0.0001t2pr1.5λ(90t+36p+45r8100)

Thus,

Ft=0.0002t2pr1.590λFp=0.0001t2pr1.536λF=r0.00015t2pr1

(b)

To determine

To calculate: The maximum number of response exceeds.

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