Question

Emerson is making a box without a top from a rectangular piece of cardboard, with dimensions 12in by 16in, by cutting out square corners with side length x in.

A) Write an equation for the volume V of the box in terms of x.

B) Use technology to estimate the value of x that gives the greatest volume. Round the value to the nearest tenth.

C) Assume Emerson used the value of d you found in Part (b) to make his box. What were the dimensions of Emerson’s box?

Answer 1

Explanation:

The volume of a rectangular box is width times length times height:

V = wlh

After the cardboard is folded, the width is 12 - 2x, the length is 16 - 2x, and the height is x.

So the volume is:

V = (12 - 2x) (16 - 2x) x

If we graph this, we get a wave: desmos.com/calculator/rsjosgzuxz

The wave is the highest at around x = 2.3 in.

If we set x = 2.3:

w = 12 - 2x = 7.4

l = 16 - 2x = 11.4

h = x = 2.3