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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.)

Respuesta :

meerkat18
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.)

The total cost C of the aquarium in terms of the height h in meters is;

C = 72√(20/h) + 36h + 200/h + 40h√(20/h)

We are given;

Volume of aquarium; V = 20 m³

Cost of glass; $10 per m²

Cost of frame; $9 per m

We are told that the base is a square. Thus;

Length = Width = L

Height = h

V = L²h

L²h = 20 m³

Thus;

L = √(20/h)

Let us solve for the metal frame:

Since there are four perpendicular edges and 8 horizontal edges, then total  length of frame is;

Frame length = 8√(20/h) + 4h

We are told frame costs $9 per m.

Thus;

Frame cost = 9(8√(20/h) + 4h])

Frame cost = 72√(20/h) + 36h

Let us solve for the glass:

Since length of base is L = √(20/h)

Area of base; A = L² = √(20/h) × √(20/h)

Area of base; A = 20/h

Area of four perpendicular side = 4h ×√(20/h)

Area of four perpendicular side = 4h√(20/h)  

Thus, cost of glass is;

10(20/h + 4h√(20/h))

⇒ 200/h + 40h√(20/h)

Total cost C is;

C = 72√(20/h) + 36h + 200/h + 40h√(20/h)

Read more at; https://brainly.com/question/14817249

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