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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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Construction An animal shelter plans to use the side of a barn as a boundary for two adjacent rectangular enclosures (see figure). Fencing for the perimeter costs $10 per foot. To separate the enclosures, a fence that costs $4 per foot will divide the region. The total area of the two enclosures is to be 6000 square feet.

(a) Find the dimensions that will minimize the cost of the fencing.

(b) What is the minimum cost?

Chapter 7.6, Problem 39E, Construction An animal shelter plans to use the side of a barn as a boundary for two adjacent

(a)

To determine

To calculate: The dimension that will minimize the cost of fencing.

Explanation

Given Information:

An animal shelter plans to use the side of a barn as a boundary for two adjacent rectangular enclosures the perimeter cost $10 per foot. Total area of the two enclosures is to be 6000 square feet.

The provided diagram 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.

Perimeter of the rectangle is:

P=2x+2y

Calculation:

As given, cost of fencing at perimeter is $10per foot.

As, P=2x+2y

Area of two adjacent rectangles is:

A=2xy

Where, 2x is length and width is y.

Total area of the two enclosures is to be 6000 square feet

(b)

To determine

To calculate: The minimum cost of fencing.

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