   Chapter 7.6, Problem 34E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Cost A manufacturer has an order for 2000 units of all-terrain vehicle tires that can be produced at two locations. Let x 1 and x 2 be the numbers of units produced at the two locations. The cost function is modeled by C   =   0.25 x 1 2 +   10 x 1 +   0.15 x 2 2 +   12 x 2 . Find the number of units that should be produced at each location to minimize the cost.

To determine

To calculate: The number of units that should be produced at each location to minimize the

cost by the manufacturer for the cost function C=0.25x21+10x1+0.15x22+12x2 where x1 and x2 are the number of units produced at two locations and 2000 units of all-terrain vehicle will be produced.

Explanation

Given Information:

The cost function C=0.25x21+10x1+0.15x22+12x2 where x1 and x2 are the number of units produced at two locations and 2000 units of all-terrain vehicle is produced.

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,

C=0.25x21+10x1+0.15x22+12x2

Since 2000 units of all-terrain vehicle is produced at two locations, where x1, x2 is the number of units, then constraint equation would be x1+x2=2000.

So, g(x,y)=x1+x22000

Now, the new function F is,

f(x1,x2,λ)=0

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