Question

A rectangular piece of cardboard measuring 10 in by 14 in is to be made into a box by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches. Answer the following questions.

Answer 1

The restrictions on x would be; 0 < x < 5.

A function V that gives the volume of the box as a function of x can be expressed as V(x) = (10 - 2x)(14 - 2x)x

The value of x whose volume would be a​ maximum is 3.2 inches

This maximum value would be 87.552 inches.

The volume will be greater than 24 cubic inches when x is less than 3.67 inches.

What are the restrictions on x?

x cannot be 0 or greater than 5, otherwise, the squares would be larger than the cardboard dimensions or no material would remain for the box, so we set the restrictions at 0 < x < 5.

A function V that gives the volume of the box as a function of x should consider the dimensions of the remaining cardboard after cutting the squares (length = 10 - 2x, width = 14 - 2x, height = x) and multiply them to find the volume.

So, the resulting function would be

V(x) = (10 - 2x)(14 - 2x)x

To find the maximum volume, we need to find the maximum value of V(x). This can be done by taking the derivative of V(x) with respect to x and setting it equal to 0:

dV(x)/dx = 4(16 - 5x)

Setting dV(x)/dx = 0, we get

64 - 20x

64 = 20x

x = 64/20

x = 16/5

= 3.2 inches