Question

A box with a square base and no top is to be built with a volume of 4000 in3. find the dimensions of the box that requires the least amount of material. how much material is required at the minimum?

Answer 1

Let the length of the square base be x inches .

And the height of the box be y inch .

Volume = x*x*y

4000 = x *x *y

$y = \frac{4000}{x^2}$

S = x*x + 2xy + 2 x y

$S = x^2 + 4 xy$

$S = x^2+ 4x* \frac{4000}{x^2}$

$S = x^2 + \frac{16000}{x}$

$S ' = 2x - \frac{16000}{x^2}$

S ' =0

$2x - \frac{16000}{x^2}=0$

$2x^3 - 16000 = 0$

$x^3 = 8000$

x = 20

$y = \frac{4000}{20^2} = \frac{4000}{400} = 10$