   Chapter 3.7, Problem 14E

Chapter
Section
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 dimensions of the box that minimize the amount of material used

Explanation

1) Concept:

i. First derivative test for absolute extreme values- suppose that c is a critical number of a continuous function f defined on an interval.

a) If f'x>0 for all x<c and f'x<0 for all x>c, then fc is the absolute maximum value of f

b) If f'x<0 for all x<c and f'x>0 for all x>c, then fc is the absolute Minimum value of f

ii. A critical number of a function f   is a number c in the domain of f  Such that either  f'c=0 or f'c does not exist.

2) Given:

Volume of the box= 32000cm3

3) Formula:

Volume of the box= x2h

4) Calculation:

Surface area of the box is proportional to the amount of material used, so we want to minimize the surface area for the given volume

Let x denote the length of the sides on the square base and let h be the height of the box

Then the Volume of the box is given by V=x2h

The given volume of the box= 32000cm332000=x2h

Divide by x2 on both sides

h=32000x2

Now, the surface area of the box is (since it has an open top) given by

A=x2+4xh

Now, put h=32000x2 in A=x2+4xh

A=x2+4x32000x2

=x2+128000x

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