# What height h and base radius r will maximize the volume of the cylinder if the container in the shape of a right circular cylinder with no top has surface area #3pi ft^2#?

##### 1 Answer

The maximum volume occurs when

**Set-Up (find the function to optimize)**

For a cylinder the volume is

And for a cylinder with no top, the surface area is

Given the area is

Solving for

(I now it will work because I've been doing this for years. But a student isn't sure.)

leads to

Substituting in the formula for volume, we get:

This is the function we've been asked to maximize.

**Optimizing the function**

**Answering the question**

Now we re-read the question to decide how to answer it. We were asked for

When

**Answer:**

The maximum volume occurs when