A manufacturer 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.

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calculista calculista · Answer 1 · 2019-11-18T16:43:05+01:00

Answer:

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

[tex]V=LWH[/tex]

where

[tex]L=(7-2x)\ in\\W=(2-2x)\ in\\H=x\ in[/tex]

substitute

[tex]V=(7-2x)(2-2x)x[/tex]

Convert to expanded form

[tex]V=(7-2x)(2-2x)x\\V=(14-14x-4x+4x^{2})x\\V=14x-14x^2-4x^2+4x^{3}\\V=4x^{3}-18x^{2} +14x[/tex]

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