   Chapter 3.7, Problem 15E

Chapter
Section
Textbook Problem

# If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

To determine

To find:

The largest possible volume of the box

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. Surface area of the box is proportional to the amount of material used

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

2) Given:

Material available =1200cm2

3) Calculation:

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

Then the Volume of the box is given as V=x2y

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

A=x2+4xy=1200cm2

Now we have x2+4xy=1200

Subtract x2 from both sides

x2+4xy-x2=1200-x2

4xy=1200-x2

Divide by 4x on both sides

y=12004x-x24x

=300x-x4

Now, put y=Vx2 in y=300x-x4Vx2=300x-x4

Now multiply by x2 on both sides,

300x-x34=V

Differentiate above equation by power rule,

300

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