I cant figure out how to do this, please help.

Question

From a 15-inch by 15-inch piece of metal, squares are cut out of the four corners so that the sides can be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are to be cut out. Express the volume of the box as a function of x.

Answer 1

Hi Kirsten.

 

So initially you have a square piece of metal, 15" x 15". You cut out squares from the corners of each side with side length x.

 

So you'll have a sheet that looks something like this now (assuming the formatting works):

 

    _____

__|       |__

|               |

|__        __|

    |____|

 

Your original side length was 15, but it has now been reduced by x on each side, so you're now going to have a new side length of 15 - 2x

 

When you fold the box up, the box will have a height of just x.

 

So you have a base with lengths (15 - 2x) and (15 - 2x), and a height which is just x.

 

V = l*w*h

 

So

 

V = (15 - 2x)(15 - 2x)(x)

 

= 225x - 60x^2 + 4x^3

 

Hope this helps!

Answer 2

A notch shaped like a square, x by x inches, is to be cut (and removed from each corner of the 15" by 15" square sheet.

When the " folding up" is completed, the box (without any top) will be formed, and its base will be a square that is (15-2x)" on each side.

 

The volume of the topless box will be is Base Area (15-2x)² times the height, x.