Maximum and minimum values?

Question

This question is confusing and I was wondering where to begin?
Its a question out a textbook.

The base of an aquarium with a given volume V is made of slate and the sides are made of glass. If slate costs five times as much(per unit area) as glass, find the dimensions of the aquarium that minimize the cost of materials.

Answer 1

So you want to minimize 5ab + 2(ac+bc) knowing that abc=V. Equivalently, you want to minimize 5/c + 2/b + 2/a . But since the product 5/c x 2/b x 2/a is 20/V, the minimum of the sum is attained when they are all equal, that is c=L/5, a=b = L/2 So L^3 =20 V and the rest follows.

Answer 2

Must the aquarium have 1 rectangular base and 4 rectangular sides?

If that's the case then ...

Volume = Width x Length x Height

The area of the base = Width x Length

The area of the sides = (2 x Width x Height) + (2 x Length x Height)

The cost of the materials is 5WL + 2WH + 2LH so that's what you are trying to minimize.

That can also be expressed as 5V/H + 2V/L + 2V/W.

I hope that helps some.

Answer 3

I feel the question may not yield correct answer. For any given volume the area of the base can be indefinitely reduced and the sides can be increased, untill the area of the base is almost zore to satisfy the minimum cost criteria. The question should be properly framed by including the total surface area of the walls and bas as additional criteria.

Answer 4

It is a calculus problem in Integration and limits, and maxima minima. V is kind of give but I think you missed telling us the shape of aquarium, most have to be rectangular tank ,but some can have square base, , or some are cube in shape. We can not have cube shape but we do have to have still shape defined. say length L, Width W, Height H by making W=H we can simplify the problem.

So why not send the actual problem by coping it.

Please re phase it.