Question

An aquarium is to be constructed to hold 2160 in3. The base is to made of slate and the sides of glass. If slate costs 5 times as much as the glass per sq in, find the dimensions that will minimize the cost of constructing the aquarium.

I started going about this problem by figuring out what is given. Height = 15 , base = 12 x 12....C = 5xy + 2 xz + 2 zy ; xyz = 2160...base is x by y

C(x,y) = 5 x y + 2 (2160) / y + 2 (2160)/x

Is this correct thus far?

Answer 1

How do we know h = 15 ?
If we do not know h, then the solution is something that looks like a space needle :)

L * w * h = 2160
if h = 15
L * w = 144
L = 144/w

cost = c = 5(L*w) + 2 L*h + 2 w*h
c = 5 L w + 2 w h + 2 L h
if we know h = 15, then
c = 5 L w + 30 w + 30 L
c = 5*144 + 30 w + 30 (144/w)
dc/dw = 0 for min
= = 0 + 30 - 30(144)/w^2
w = 12