   Chapter 4.7, Problem 15E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 maximum volume of a box made from 1200 cm2 of material with a square base and an open top.

Explanation

Given:

The surface area of the given material 1200cm2 .

Calculation:

Let the side of the square base be x cm and the height of the box be y cm.

From Figure 1, the area of the square base is x2 cm2 .

The area of each side of the box is xy cm2 .

Therefore, the total surface area of the given box is x2+4xy .

x2+4xy=12004xy=1200x2y=300xx4

The volume of the box is V=x2y cm2 .

Substitute y=300xx4 in A=x2y ,

V=x2(300xx4)=300xx34

Differentiate V with respect to x,

dVdx=3003x24

For critical points, dVdx=0 .

3003x24=03x24=300x2=400x=±20

x is the length of the box, and so it cannot be negative

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