A square has sides of 18 units. Squares x + 1 by x + 1 units are cut out of each corner, and then the sides are folded up to create an open box.

Question

A square has sides of 18 units. Squares
x + 1
by
x + 1
units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume V of the box as a function in terms of x.
V(x) =

Answer 1

For problems like this I always start by making a sketch (tough to do on this forum).

 

I'd draw the original square, carving out the (x+1)by(x+1) corners.  Then visualize folding the corners up into the rectangular prism (box).

 

So what are the dimensions for this box.

 

Length=18-(x+1)-(x+1)=18-2(x+1)=18-2x-2=16-2x=2(8-x)

Width=2(10-x)  <-- Same as the length since we are starting with a cube 

Height=x-1

 

The volume of the box is simply (Length)(Width)(Height)

 

Substituting our expressions for each of these dimensions we have

 

Volume=2(8-x)2(8-x)(x+1)

=4(64-16x+x²)(x+1)

=4(64x-16x²+x³+64-16x+x²)

=4(x³-15x²+48x+64)

=4x³-60x²+192x+256

 

Good Luck

 

Cheers

 

-Chris