Find dimensions for the least cost?

Question

A box of volume 384 cubic feet having a square base and no top is to be constructed from material costing $3 per square foot for the base, and $1 per square foot for the sides. Find the dimensions that would cost the least.

Answer 1

Cost of base = $3 per square foot
Cost of Side = $1 per square foot
There is no top for box.
Suppose side = x
height = h
Volume of box = 384 cubic feet
(x^2)h = 384
h = 384/x^2
Surface Area A = x^2 + 4xh
A = x^2 + 4x(384/x^2)
A = x^2 + 1536/x
Cost , C = 3*x^2 + 1*1536/x
C = 3x^2 + 1536/x
dC/dx = 6x - 1536/x^2 = 0
x^3 = 256
x = 6.3496
h = 384/(6.3496)^2
h = 9.5244

Base Area = 6.3496*6.3496 = 40.317 square foot
Side Area = 4*9.5244*6.3496 = 241.905 square foot
Least cost = 40.317*3 +241.905*1 = $362.856

Answer 2

if side of the base is x feet and height is y feet

volume = 384
implies x^2 .y = 384 or y = 384 / { x^2 }


base area = x^2
lateral surface area = 4xy

cost , C= 3 x^2 + 1*(4xy)

eliminate x or y using x^2 .y = 384

eliminating y using y = 384 / { x^2 }

C = 3 x^2 +1536/x

now do ordinary minimisation