Question

An open rectangular tank is made so as to contain 4.5 metres cube of liquid. The length must be twice the breadth and the amount of material used to make the tank must be minimum. Find the dimensions of the tank.

Answer 1

Volume is given to be 4.5 m^3

v = length x width x height
v = lwh
want everything in one variable w
given that;
l = 2w
using volume...
h = 4.5 / lw
h = 4.5 / 2w^2

surface area = wl + 2hl + 2wh
sub for length and height
s = 2w^2 + 9/w + 4.5/w
s = 2w^2 + 13.5/w
derivative s
s' = 4w - 13.5/w^2
0 = 4w - 13.5/w^2
13.5/w^2 = 4w
13.5 = 4w^3
13.5/4 = w^3
3.375 = w^3
cube root
1.5 meters = w
then the length = 2(1.5) = 3 meters
height = 4.5/(3*1.5) = 1 meter