# Derivative problem

#### 39mello

##### New member
A glass shaper wants to build a cylindrical-shaped glass, but has enough raw material for just 48π cm2 of glass. How much should be the base radius and the
height for this cup to have as much volume as possible? What's the biggest volume you can get in this situation?

(Note: remember that this cup doesn't have a lid.)

#### lev888

##### Elite Member
A glass shaper wants to build a cylindrical-shaped glass, but has enough raw material for just 48π cm2 of glass. How much should be the base radius and the
height for this cup to have as much volume as possible? What's the biggest volume you can get in this situation?

(Note: remember that this cup doesn't have a lid.)
It appears you didn't read the posting guidelines.

#### Otis

##### Elite Member
Hi 39mello. It's difficult to help students online when they don't explain why they're stuck. Please show us how far you got.

Here are my beginning thoughts. The inside volume of the cup is the cylinder you're working with, not the cup itself. In other words, we can ignore the thickness of the glass. The amount of glass is actually stated in terms of the inside cylinder's surface area (square centimeters).

Before writing the volume function to maximize, you first need to determine an expression for the cylinder's height in terms of its radius (because we want a function of one variable, not two). For that, we could use the surface area formula for a cylinder, adjusted for no top. Substitute the given surface area into that formula, and solve the equation for h. That will give you the expression for h in terms of r needed for the volume function.

Please show us some work or explain where you're stuck.

#### Otis

##### Elite Member
Hi 39mello. Are you having trouble getting started?

Again, the inside volume of the cup is a cylinder. Let's think about that cylinder's surface area (no top). Two areas comprise that surface: one is circular (the base), and the other is rectangular (the wall).

Let's define the variables needed to express those areas:

h = the cylinder's height

The area of the circular base is expressed as Pi*r^2.

If we mentally unroll a cylinder's wall, we see that it's a rectangle. The width of that rectangle is the same as the circumference of the circular base, and its height is h. The circumference of a circle is 2*Pi*r. So, the area of the cylinder's wall (width×height) is expressed as 2*Pi*r*h.

Therefore, a formula for our no-top cylinder's total surface area (A) is:

A = Pi*r^2 + 2*Pi*r*h

We need an expression for h in terms of r, for the volume function. We can obtain that expression, by substituting the given area into the formula above and solving for h.

48*Pi = Pi*r^2 + 2*Pi*r*h

Have a go at it, and post what you get for h. If you're able to write the cylinder's volume function V(r), then please share that, too. Cheers!

#### Otis

##### Elite Member
For anyone interested, we have the following results.

h = -r/2 + 24/r

The volume of a right cylinder is the base area times the cylinder's height.

V(r) = -1/2 Pi r^3 + 24 Pi r

The first derivative's critical value is the radius for which the volume is maximized.

V'(r) = -3/2 Pi r^2 + 24 Pi

0 = -3/2 Pi r^2 + 24 Pi

r = 4

h = -4/2 + 24/4 = 4

The cylinder volume will be maximized when the radius and height are each 4 centimeters.

V(4) = -1/2 Pi (4^3) + 24 Pi (4) = 64(Pi)

The maximum volume is 201 cubic centimeters (rounded).