# Math Help - Cola Can Project

1. ## Cola Can Project

In my Ap Caluculus class we are on the chapter of derivative applications and we have this project to di.

A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)

If someone could please help to get me started on this problem I would be very grateful.

2. Originally Posted by Vigo
In my Ap Caluculus class we are on the chapter of derivative applications and we have this project to di.

A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)

If someone could please help to get me started on this problem I would be very grateful.
The amount of material required is proportional to the
surface area of the cylinder. This is:

$Area\ =\ (2.\pi.r).h\ +\ 2.(\pi.r^2)$

that is the area of the curved part of the can, plus the area of the
two end pieces.

The volume of the cylinder is:

$Volume\ =\ \pi.r^2.h$.

Where $r$ is the radius of the base, and $h$ is the height of
the can.

That the volume is fixed allows you to write $h$ as a function
of $r$ by rearranging the second of these equations. Substituting
this for $h$ in the first equation gives you the surface area
as a function of $r$ and $Volume$.

You then need to find the minimum of $Area$ by
differentiating w.r.t. $r$, and setting the derivative
equal to $0$.

RonL

3. OK so:

h = 12/(pi*r^2)

and

f(x) = (2*pi*r)*(12/pi*r^2) + 2(pi*r^2)

Find the derivative of f(x) and set that equal to 0.

Is all of this right?
If it is, what does the final answer tell you?
And where does the 1 fl. oz. = 1.80469 inches cubed come into this problem?
Thanks again.

4. Originally Posted by Vigo
OK so:

h = 12/(pi*r^2)

and

f(x) = (2*pi*r)*(12/pi*r^2) + 2(pi*r^2)

Find the derivative of f(x) and set that equal to 0.

Is all of this right?
If it is, what does the final answer tell you?
And where does the 1 fl. oz. = 1.80469 inches cubed come into this problem?
Thanks again.
If you use inches as you unit of linear measure (i.e. for h and r), then
you should use cubic inches as your unit of volume. So you should
have:

h = (12*1.80469)/(pi*r^2),

RonL

5. Alright thanks a lot for all of the help. I am going to work on this and post my final answers soon.