A manufacturer cuts squares from the corners of an 8 cm by 14 cm piece of sheet metal and then folds the metal to make an open-top box.

Question

Show that the volume of the box is:

V(x)=x(8-2x)(14-2x)

Answer 1

The volume of a rectangular prism is length * width * height.

 

Since the dimensions of the cut out square are unknown, they have an edge length of x. 

 

The original length (14 cm) reduces by x on the left, and another on the right. So L = 14 - 2x.

 

The original width (8cm) is reduced by x on the front and the back. So W = 8 - 2x

 

The original height was zero, since sheet metal is flat. When the x by x squares were cut out, the corners, four "flaps" were produced. These larger of these flaps were rectangles with dimensions x and 14 - 2x, and the smaller had dimensions x and 8 - 2x. When these flaps were folded upward, the shape took on a third dimension of height, with a value x. 

 

So V =L*W*H

Or V = (14-2x)(8-2x)x