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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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Cost A manufacturer has an order for 1000 units of fine paper 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 +   25 x 1 +   0.05 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 for cost function C=0.25x12+25x1+0.05x22+12x2 where 1000 units of fine paper can be produced at two locations and x1, x2 is assumed to be the number of units that should be produced at each location.

Explanation

Given Information:

The provided cost function is C=0.25x12+25x1+0.05x22+12x2 where 1000 units of fine paper can be produced at two locations, and x1, x2 is assumed to be the number of units that should be produced at each location.

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=0.25x12+25x1+0.05x22+12x2

Since 1000 units of fine paper can be produced at two locations, where x1, x2 is the number of units, then constraint equation would be x1+x2=1000.

So, g(x,y)=x1+x21000

Now, the new function F is,

f(x,y,λ)=0.25x12+25x1+0

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