Question

A box lid is to be made from a rectangular piece of cardboard measuring 72 cm by 216 cm. Two equal squares of side x are to be removed from one end, and two equal rectangles are to be removed from the other end so that the tabs can be folded to form a box with a lid. Find x such that the volume of the box is a maximim

Answer 1

have not seen that variation of the popular "make a box" question before.
Made a diagram showing the net of the box with a lid.

let the sides of the square to be cut out be x cm
the the width of the box is 72 - 2x
let the length of the box be y cm
According to my sketch,
2x + 2y = 216
x+y = 108
y = 108-x
the height of the box is x

Volume = V = lwh = (72 - 2x)(108-x)(x)
= x(7776 - 288x - 2x^2)
= 7776x - 288x^2 - 2x^3
dV/dx = 7776 - 576x - 6x^2
= 0 for a max of V
x^2 + 96x - 1296 = 0
(x - 12)(x + 108) = 0
x = 12 or x = a negative, which is silly

so x = 12

Answer 2

wrong