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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.

To determine

To find: The dimension of the box that minimize the amount of material used whose volume is 3200 cm3 .

Explanation

Given:

The volume of a box with square base and open top = 3200 cm3 .

Calculation:

The volume of a box with a square base x by x cm and height h cm is V=x2h .

The amount of material used is directly proportional to the surface area. So, we will minimize the amount of material by minimizing the surface area.

The surface area of the box is A=x2+4xh .

So, V=x2×h=32000cm3 .

This provides the result as, h=32000x2 .

Therefore, area becomes (substitute the value of h)

A=x2+4x(32000x2)=x2+128000x

Differentiate A with respect to x,

dAdx=2x128000x2

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