Amanufacturer cuts squares from the corners of a rectangular plece of sheet metal that measures 2 inches by 7 inches (see figure 1). the manufacturer then
folds the metal upward to make an open-topped box (see figure 2). letting x represent the side-lengths (in inches) of the squares, use the aleks graphing
calculator to find the value of x that maximizes the volume enclosed by this box. then give the maximum volume. round your responses to two decimal places.

The value of x that maximizes the volume enclosed by this box is 0.46 inches

The maximum volume is 3.02 cubic inches

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The volume of the open-topped box is equal to

where

substitute

Convert to expanded form

using a graphing tool

Graph the cubic equation

Remember that

The domain for x is the interval -----> (0,1)

Because

If x>1

then

the width is negative (W=2-2x)

so

The maximum is the point (0.46,3.02)

see the attached figure

therefore

The value of x that maximizes the volume enclosed by this box is 0.46 inches

The maximum volume is 3.02 cubic inches