Question

An aquarium with a square base has no top. There is a metal frame. Glass costs 7 dollars/m² and the frame costs 10 dollars/m. The volume is to be 20 m³.

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

Let x be the length of each side of the square base, and let h be the height of the aquarium. Then the volume of the aquarium is:

V = x^2 * h

We are given that V = 20, so we can solve for x:

x^2 * h = 20

x^2 = 20/h

x = sqrt(20/h)

The area of each of the four walls of the aquarium is x*h, and there are four walls, so the total area of glass needed is:

A_glass = 4xh = 4sqrt(20h)

The cost of the glass is 7 dollars/m², so the cost of the glass is:

C_glass = 7A_glass = 74sqrt(20h) = 28sqrt(20h)

The perimeter of the square base is 4x, so the length of the metal frame needed is:

L_frame = 4x + 4h

Substituting x = sqrt(20/h), we get:

L_frame = 4sqrt(20/h) + 4h

The cost of the frame is 10 dollars/m, so the cost of the frame is:

C_frame = 10*L_frame = 10(4sqrt(20/h) + 4h)

The total cost is the sum of the cost of the glass and the cost of the frame:

C = C_glass + C_frame = 28sqrt(20h) + 10(4sqrt(20/h) + 4h)

Simplifying, we get:

C = 28sqrt(5h) + 40sqrt(5h) + 40h

C = 68sqrt(5h) + 40h

Therefore, the total cost C is a function of the height h and can be expressed as C = 68sqrt(5h) + 40h in dollars.

Explanation: