A box with an open top and a square base is to hve a volume of 108 cubic inches. What dimensions allow for the least amount of material o construct the box?
Calculus

Question

A box with an open top and a square base is to hve a volume of 108 cubic inches. What dimensions allow for the least amount of material o construct the box?
Calculus

anonymous · Answer

base's side: x
height: h

Volume: hx^2 = 108
h = 108/x^2

Surface area: 4hx + x^2 = x^2 + 432/x

anonymous · Answer

tell me if you don't understand anything

surface area  f(x) = x^2 + 432/x
to find the minimum of this, we need to take the first derivative

f'(x) = 2x - 432/x^2
then set f'(x) = 0

2x - 432/x^2 = 0
2x^3 - 432 = 0
x^3 = 216
x = 6

h = 108/x^2 = 108/36 = 3

anonymous · Answer

got it @npowers8 ?

anonymous · Answer

yes I am pretty sure. so the dimensions are 6 inches for base and 3 inches high

anonymous · Answer

yes