An open cardboard box is to be made by cutting squares of side x cm from each corner of a card of side 60cm and folding the resulting "flaps" up to form the box. Find the value of x that gives the box a maximum capacity.

Answer 1

dimension of the box is
60-2x by 60-2x by x , where x < 30

Vol = x(60-2x)^2
= 3600x - 240x^2 + 4x^3
d(Vol)/dx = 3600 - 480x + 12x^2
= 0 for a max/min of Vol

divide each term by 12
x^2 - 40x + 300 = 0
(x-10)(x-40) = 0
x = 10 or x = 40, but x<30

x = 10

cut out flaps of 10 cm by 10 cm

check:
if x = 10 , Vol = 10(40)^2 = 16000
if x = 10.1 , Vol = 10.1(39.8)^2 = 15998.8 which is < 16000
8
if x = 9.9 , Vol = 9.9(40.2)^2 = 15998.8 which is < 16000
My answer is very plausible