# How do you find the dimensions of the aquarium that minimize the cost of the materials if the base of an aquarium with volume v is made of slate and the sides are made of glass and the slate costs five times as much (per unit area) as glass?

##### 1 Answer

Without a specific value for the volume,

We need to develop an expression for the cost of materials based on one of the dimensions.

Let

Note that the minimum will be achieved when the width and the depth are equal. (I think this should be obvious but re-post if it isn't).

So we are only interested in the case (substituting

The volume in this case can be expressed as

So if we re-write our Cost equation as

This becomes

To find the minimum cost, we take the derivative of this and set the result to zero.

Once a value has been established for the volume (