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### Design a More Efficient Soda Can

```
Date: 23 Apr 1995 21:54:38 -0400
From: Ryan R. Bullock
Subject: Geometry Problem

I have a problem for geometry that I need help with. Basically, the problem
is to design a more efficent soda can that holds the regular 12oz (355 Ml.)
of liquid. The can needs to have the least possible surface area. It would
be easy to do this on a graphics calculator, but to find the area of the
side of the can, you need two varibles (Circumference and height). Any help
would be greatly appreciated. Thank You.

Ryan R. Bullock,
```

```
Date: 23 Apr 1995 22:20:04 -0400
From: Dr. Ken
Subject: Re: Geometry Problem

Hello there!

Is the can going to retain the shape of a cylinder?  If it doesn't have to,
the problem becomes a whole new ball-game; the answer is well-known to be a
sphere, but proving that is another matter.

-K
```

```
Date: 24 Apr 1995 20:07:43 -0400
From: Ryan R. Bullock
Subject: Re: Geometry Problem

The can should retain the shape of a cylinder. The can needs to hold 355ml
of liquid, but I need to find the dimensions of the can that would hold this
volume with the least possible surface area. Thanks for your help!

Ryan R. Bullock,
```

```
Date: 25 Apr 1995 23:51:44 -0400
From: Dr. Ken
Subject: Re: Geometry Problem

Hello!

Well, this is one of the classic Max/Min problems, and it's ideally suited
for calculus or a graphing calculator.  I think I'll just give you a hint
first, and please write back if you're still having trouble.

Since we know that the can must retain its volume of 355 ml, we have an
equation in the radius, r, of the can and its height, h:  h*Pi*r^2 = 355.
Therefore, we can solve for h, and we get h = 355/(Pi*r^2).

Now we need to get a formula for the surface area of the can.  The can has
three faces (the side panel, the top and the bottom), and the area of each
face depends only on r and h.  See if you can find an expression for the
total surface area of the can in terms of r and h.

Then comes the cool part.  You're going to need to get rid of one of the
variables, because equations with two variables in them can be nasty.  You
can do that, because you've got that great equation that tells you
what h is in terms of r, so you can substitute that into the formula for
surface area.  Then, VOILA!  You'll have a formula for surface area that
only depends on the radius of the can.

As I said before, please write back if you still can't figure it out, and
we'll give you a bigger hint.

-K
```
Associated Topics:
High School Calculus

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