Question
A box has a Square base. The Perimeter of the base plus the height is 120cm. What is the max volume of this box, and what are the dimensions of the this maximized box.
Answers
Reiny
let the base be x by x cm
let the height be y
given: 4x + y = 120
y = 120 - 4x
V = x^2 y = x^2(120-4x)
= 120x^2 - 4x^3
dV/dx = 240x - 12x^2 = 0 for a max of V
12x^2 - 240x = 0
12x(x- 20) = 0
x = 0 ----> yields a minimum volume
or
x = 20
so the box is 20 cm by 20 cm by 40 cm for a max volume of 16000 cm^3
let the height be y
given: 4x + y = 120
y = 120 - 4x
V = x^2 y = x^2(120-4x)
= 120x^2 - 4x^3
dV/dx = 240x - 12x^2 = 0 for a max of V
12x^2 - 240x = 0
12x(x- 20) = 0
x = 0 ----> yields a minimum volume
or
x = 20
so the box is 20 cm by 20 cm by 40 cm for a max volume of 16000 cm^3