Question

we want to construct a box with a square base and no top. the volume of the box will be 8 cm^3 what are the dimension of the minimize the surface area .

Answer 1

let base be x by x
let the height be y
x^2 y = 8
y = 8/x^2

SA = base + 4 sides
= x^2 + 4xy
= x^2 + 4(8/x^2)
= x^2 + 32/x
d(SA)/dx = 2x - 32/x^2
= 0 for a minimum of SA

2x = 32/x^2
2x^3 = 32
x^3 = 16
x = cuberoot(16) = appr 2.52
then y = 1.26

the base is 2.52 by 2.52 and the height is 126

(notice the height is half of the side of the base)