Question

An aquarium with a square base has no top. There is a metal frame. Glass costs 8 dollars/m^2 and the frame costs 7 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

Given that the volume of the aquarium is 20m^3.

Volume = Area of Base x height

Area of Base = Volume / height = 20/h

Given that the aquarium has a square base.

Area of square = l^2

Thus, the length of the base of the aquarium is
`√(area \ of \ base) = \sqrt{ (20)/(h) }`

The frame is to cover 8 sides with the length equal to the length of the base and 4 sides with the length of the height.

Thus, the total perimeter of the frame is given by
`8\sqrt{(20)/(h)}+4h= \sqrt{64\left((20)/(h)\right)}+4h = \sqrt{(1,280)/(h)}+4h`

Area of the four side faces of the aquarium is 4 times the length of the base times the height =
`4*\sqrt{ (20)/(h) }* h=\sqrt{16\left((20)/(h)\right)h^2}=√(320h)`

Total area to be covered by grass is the base and the four side faces and is given by
`(20)/(h)+√(320h)`

Cost of the entire metal frame =
`7\left(\sqrt{(1,280)/(h)}+4h\right)= \sqrt{49\left((1,280)/(h)\right)}+7(4h) = \sqrt{(62,720)/(h)}+28h`

Cost of the entire grass =
`8\left((20)/(h)+√(320h)\right)=(160)/(h)+√(64(320h))=(160)/(h)+\sqrt{20,480h`

Therefore, total cost in terms of the height, h, is given by


`C=\sqrt{(62,720)/(h)}+28h+(160)/(h)+\sqrt{20,480h`