The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs seven times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.)

Step-by-step explanation:

This is a minimization problem.

For this case we assume that we have a box and the volume is given by:

  (1)

For this case we know that slate costs seven times as much (per unit area) as glass so then 7xy this value and if we find the cost function like this:

If we solve z from equation (1) we got:

  (2)

So then we can replace equation (2) into the cost equation and we got:

And with this we have a function in terms of two variables x and y.

We can simplify the last equation and we got:

In order to solve the problem for the dimensions we can take the partial derivates respect to x and y and we got:

We can set the last two equations equal since are equal to 0 and we got:

And the only possible solution for this case is

So then if we use x=y for the partial derivate of x we have:

And solving for x we got:

And analogous we can do the same thing for the partial derivate of y and we got:

And solving for x we got:

And for z we can replace and we got:

So then the dimensions in order to minimize the cost would be: