Abox with a volume of 5 m3 is to be constructed with a gold-plated top, silver-plated bottom, and copper-plated sides. if gold plate costs $110 per square meter, silver plate costs $60 per square meter, and copper plate costs $17 per square meter, find the dimensions that will minimize the cost of the materials for the box

  V (Volume) = s^2 * h = 48 > h = 48s^-2 
C (Cost) = (4sh)(2) + (2s^2)(1) = 2s^2 + 8sh = 2s^2 + 8s(48s^-2) = 2s^2 + 384s^-1 
dC/ds = 4s - 384s^-2 = 0 > 4s(1 - 96s^-3) = 0 
Since s can't be 0, the only possible solution is: 
1 - 96s^-3 = 0 
s^3 - 96 = 0 
s = cube root(96) = 2 cube root(12) ~ 4.579 cm 
s^2 = 4 + 2 cube root(18) = 2[2 + cube root(18)] 
h = 48/s^2 = 24/[2 + cube root(18)] ~ 5.194 cm

 V (Volume) = s^2 * h = 48 > h = 48s^-2 C (Cost) = (4sh)(2) + (2s^2)(1) = 2s^2 + 8sh = 2s^2 + 8s(48s^-2) = 2s^2 + 384s^-1 dC/ds = 4s - 384s^-2 = 0 > 4s(1 - 96s^-3) = 0 Since s can't be 0, the only possible solution is: 1 - 96s^-3 = 0 s^3 - 96 = 0 s = cube root(96) = 2 cube root(12) ~ 4.579 cm s^2 = 4 + 2 cube root(18) = 2[2 + cube root(18)] h = 48/s^2 = 24/[2 + cube root(18)] ~ 5.194 cm