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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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Office Space Partitions will be used in an office to form four equal work areas with a total area of 360 square feet (see figure). The partitions that are x feet long cost $100 per foot and the partitions that are y feet long cost $120 per foot.

(a) Find the dimensions x and y that will minimize the cost of the partitions.

(b) What is the minimum cost?

Chapter 7.6, Problem 40E, Office Space Partitions will be used in an office to form four equal work areas with a total area of

(a)

To determine

To calculate: The dimension x and y that will minimize the cost of the partitions.

Explanation

Given Information:

Partitions will be used in an office to form four equal work areas with a total area of 360 square feet. the partitions that are x feet long cost $100 per foot and the partitions that are y feet long cost $120 per foot and the provided figure is:

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:

As given, total area of 360 square feet. the partitions that are x feet long cost $100 per foot and the partitions that are y feet long cost $120 per foot

(b)

To determine

To calculate: The minimum cost of fencing.

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