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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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The production function for a manufacturer is given by

f ( x , y ) = 60 x 0.7 y 0.3 where x is the number of units of labor (at $42 per unit) and y is the number of units of capital (at $144 per unit). The total cost for labor and capital cannot exceed $240,000. Find the maximum production level for this manufacturer.

To determine

To calculate: The maximum production level for the manufacturer with production function.

f(x,y)=60x0.7y0.3.

Explanation

Given Information:

Production function,

f(x,y)=60x0.7y0.3

Where x is the number of units of labours ($ 42 units).

And, y is the numbers of units of capital ($144 units).

42x+144y240,000

And the total cost of labours and capital can’t exceed $240,000.

42x+144y=240,0007x+24y=40000

Formula used:

LaGrange’s multipliers:

F(x,y,λ)=f(x,y)λ(g(x,y)k)

Where F(x,y,λ) is the Lagrange function.

f(x,y) is the gain function.

λ(g(x,y)k) is the constrain equation.

Where λ is the parameter constant.

Fx=Fx,Fy=Fy,Fλ=Fλ.

From langrage equation.

Find Fx=0,Fy=0andFλ=0.

Use Fx=0,Fy=0 to find the value of λ in terms of x and y and then satisfy in Fλ=0.

Calculation:

Consider the primary equation and substitute the value

F(x,y,λ)=60x0.7y0.3λ(7x+24y40000)Fx=60(0

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