An open rectangular box is to be made by cutting equal squares

Question

An open rectangular box is to be made by cutting equal squares from the corners of a square piece of cardboard measuring 18"x 18" and then folding up the cardboard to make the sides of the box. Find the dimensions (l*w*h) of the box of maximum capacity obtainable this way.

Answer 1

Let x = the length of the square cut out of each corner.  Then the length/width of our box is 18-2x.  To find volume we use LxWxH or:

 

(18-x)(18-x)(x) = x³ -36x² + 324x  Take the first derivative and set it equal to zero to mind the critical values:

 

3x² -72x + 324 = x² -24x +108 = (x-18)(x-6) = 0

 

Our critical values are 18 and 6.  Clearly 18 will not work so the answer is 6.  so our dimensions are 6 x 6 x 6

Answer 2

Draw and label a diagram!

The bottom of the box is (18 - 2x) by (18 -2x)

The height of the box is x.

A = x(18-x)(18-x)

Put into vertex form to determine maximum.