Find polynomials that represent the volume and outside surface area of the box.

Question

Use geometry to solve the problem.

A square piece of cardboard is 17 inches on a side. A square piece x inches on a side is cut out from each corner. The flaps are then turned up to form an open box. Find polynomials that represent the volume and outside surface area of the box.

 

Simplify your answers completely.

 

Volume:

 

Surface Area:

Answer 1

The initial surface area of the box is 17×17 = 289 square inches. One cut-out-square piece's area is x×x = x² square inches. Since you're cutting out four of them, one for each corner, you're cutting out a total of 4x² square inches. That means the new box's outside surface area is:

 

SA_O = 289 - 4x² square inches

 

 

For volume, we'll need to figure out the length, width, and height of the new box. Height is just x, since that's the width of the part being folded up. For length, on either side, we're losing the width of one folded-up part. That means we're losing a total length of 2x. So, length is 17 - 2x. Width is actually the same. We started with 17 inches and then lost x on each side for a total of 17 - 2x.

 

That means the new box's volume is:

 

V = length×width×height

V = (17-2x)(17-2x)(x)

V = (289-34x-34x+4x²)(x)

V = (289-68x+4x²)(x)

V = 289x - 68x² + 4x³ cubic inches