Chapter 7.5, Problem 42E

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Profit A corporation manufactures candles at two locations. The cost of producing x 1 units at Location 1 is C 1 =   0.02 x 1 2 +   4 x 1 +   500 and the cost of producing x 2 units at Location 2 is C 2 =   0.05 x 2 2 +   4 x 2 +   275 . The candles sell for $15 per unit. Find the quantity that should be produced at each location to maximize the profit P = 15 ( x 1 + x 2 ) − C 1 − C 2 . To determine To calculate: The quantity which must be produced at each location to maximize the profit P=15(x1+x2)C1C2 if the cost to produce x1 number of units at location 1 is C1=0.02x12+4x1+500 and the cost to produce x2 number of units at location 2 is C2=0.05x22+4x2+275. The candles are sold for$15 per unit.

Explanation

Given Information:

The profit is provided by P=15(x1+x2)C1C2. The cost to produce x1 number of units at location 1 is C1=0.02x12+4x1+500 and the cost to produce x2 number of units at location 2 is C2=0.05x22+4x2+275. The candles are sold for \$15 per unit.

Formula used:

If the function y=f(x) has a maximum at a point then their first partial derivatives fx(x) and fy(x) at that point are zero.

Calculation:

Consider the profit, P=15(x1+x2)C1C2, the cost to produce x1 number of units at location 1 is C1=0.02x12+4x1+500 and the cost to produce x2 number of units at location 2 is C2=0.05x22+4x2+275.

Substitute C1=0.02x12+4x1+500 and C2=0.05x22+4x2+275 in the equation P=15(x1+x2)C1C2,

P=15(x1+x2)[0.02x12+4x1+500][0.05x22+4x2+275]=15x1+15x20

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