I don’t know where to begin

Question

A box with a hinged lid is to be made out of a rectangular piece of cardboard that measures 4

 inches by 9

 inches. Six squares will be cut from the cardboard: one square will be cut from each of the corners, and one square will be cut from the middle of each of the 9

-inch sides (see Figure 1). The remaining cardboard will be folded to form the box and its lid (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.

Answer 1

So, I'm guessing Figure 1 looks like this:

Since we are being asked to calculate the maximum volume, we need an equation for the volume,

The width of the bottom would be "4 - 2x". The length of the bottom would be 9/2 - x - x/2 or "4.5 - 1.5x". The height would be "x".

So the volume is:

V(x) = x(4 - 2x)(4.5 - 1.5x)

Graphing this shows a max at x = 0.7847 (with a volume of 6.3378). Therefore the value of x that maximizes the volume (to 2 decimal places) is x = 0.78 inches and the maximum volume is 6.34 cubic inches.