An open box is made from a rectangular piece of cardboard measuring 16 cm by 10cm. Four equal squares are to be cut from each corner and flaps folded up.

Question

Find the length of the side of the square which makes the volume of the box as large as possible.
Find the largest volume

Answer 1

the base of the box is 16-2x by 10-2x, and the height is x

v = x(16-2x)(10-2x)
v'= 4(x-2)(3x-20)
20/3 >5, so it's too large
so, max volume occurs when x=2

The box is 6x12x2 with volume 144

Answer 2

@steve
Why is the base of the box 16-2x by 10-2x
(where did the 2x come from?)

Answer 3

because x was cut off each corner. Draw a diagram, and you will see that when the corners are turned up, the sides have 2x cut off.