I can't figure out how to do this, please help.

Question

From a 40-inch by 40-inch piece of metal, squares are cut out of the four corners so that the sides can be then folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of x.

Answer 1

The best way to tackle this is to draw a picture - I wish I could upload one, but because I can't I will do my best to explain it.

 

Draw a large box and label each side 40" for length, then draw a smaller box at each corner to represent the smaller squares that will be cut out. Label the sides of the smaller boxes "x". Since the smaller boxes will be cut out, we can express both the length and width of the box that will be made once folded as 40-2x (since each side of the big box will have 2 sides from 2 of the smaller boxes).

 

volume of box = length x width x height

                     =(40-2x) x (40-2x) x height

                     =(40-2x)²              x height

 

Now let's figure out how the height will be represented by looking at our picture. (I WISH I COULD UPLOAD A PICTURE, UGHHH lol). If we conceptualize folding our picture into a box, the height of the box would simply be the value of x.

 

 

volume of box = (40-2x)²x

 

So, expressed as a function, f(x) = x(40-2x)²