Question

An aquarium with a square base has no top. There is a metal frame. Glass costs 10 dollars/m^2 and the frame costs 9 dollars/m. The volume is to be 20 m^3. Express the total cost C in terms of the height h in meters. (Hint: work out the cost of the glass and frame separately.)

Answer 1

The volume is given as V = Lwh = 20 m^3.
Since the base is a square then V = L²h = 20 m^3
Rearranging, we have L = √(20/h).

For the metal frame:
8√(20/h) + 4h ==> [8 is multiplied to √(20/h) because there are 8 sides measuring L, and 4 is multiplied to h because there are 4 sides that have measure h.]

9[8√(20/h) + 4h]
=72√(20/h) + 36h = Cost of metal frame

For the glass:
Area of one square base = L² = 20/h
Area of one rectangular side = h√(20/h) 
Area of 4 rectangular sides = 4h√(20/h) 

10[20/h + 4h√(20/h)] = Cost of glass

C = 72√(20/h) + 36h + 10[20/h + 4h√(20/h)] 

(take note, we did not multiply 10 to the terms because 10 dollars is per unit area and not unit length.)