You are given a rectangular sheet of cardboard that measures 11 in. by 8.5 in. (see the diagram below). A small square of the same size is cut from each corner, and each side folded up along the cuts to from a box with no lid. 1. Anya thinks the cut should be 1.5 inches to create the greatest volume, while Terrence thinks it should be 3 inches. Explain how both students can determine the formula for the volume of the box. Determine which student's suggestion would create the larger volume. Explain how there can be two different volumes when each student starts with the same size cardboard. 2. Why is the value of x limited to 0 in. < x < 4.25 in.?

We can solve for the value of x using the formula:

V = l w h

where,

h = x the size of the cut since it would form the walls of the rectangle

w = 8.5 - 2x = it is subtracted by 2x since two sides will be cut

l = 11 - 2x

Substituting:

V = x (8.5 - 2x) (11 - 2x)

Expanding the expression:

V = 93.5 x - 39 x^2 + 4 x^3

To solve the maxima, we have to get the 1st derivative dV / dx then equate to 0. dV / dx = 0:

dV / dx = 93.5 - 78 x + 12 x^2

0 = 93.5 - 78 x + 12 x^2

We get:

x ≈ 1.585 in and x ≈ 4.915 in

Therefore Anya’s suggestion of 1.5 inches would create the larger volume since it is nearer to 1.585 inches.

There can be different volumes since volume refers to the amount of space inside the rectangle. They can only have similar perimeter and surface area, but not volume.

It is restricted to 0 in. < x < 4.25 in. because our w is 8.5 - 2x. Going beyond that value will give negative dimensions.