# Cola Can Project

• Dec 6th 2005, 02:00 PM
Vigo
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.
• Dec 6th 2005, 07:50 PM
CaptainBlack
Quote:

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:

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

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:

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

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

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

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

RonL
• Dec 7th 2005, 11:19 AM
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.
• Dec 7th 2005, 12:06 PM
CaptainBlack
Quote:

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
• Dec 8th 2005, 07:11 PM
Vigo
Alright thanks a lot for all of the help. I am going to work on this and post my final answers soon.