Units and Cylinder Volume

Date: 02/06/2003 at 20:59:10
From: Dan
Subject: Cylinders

My 7th-grade daughter brought home a question:

A cylindical storage tank has a radius of 15ft and a height of 30ft.
Find the volume of the tank.

    to find volume: Pi x r^2 x H

My answer was

   21205.75

It was wrong.  I am totally lost. I know I used the correct formula
but can't figure out what could be wrong.

Thanks,
A Very Lost Dad in the new world of math.

Date: 02/06/2003 at 22:36:15
From: Doctor Ian
Subject: Re: Cylinders

Hi Dan,

>Pi x r^2 x H

This is the right formula.  So let's see what we get:

  volume = pi * r^2 * h

         = pi * (15 ft)^2 * 30 ft

         = pi * 15 * 15 * 30 ft^3
 
         = pi * 6750 ft^3

Now, in many cases, this is as far as you have to go, but if you want
to end up with a number, you have to multiply by pi.  The most common
value for pi is 3.14.  I like fractions, so I like to use 22/7. 
They're both approximations, and both about equally good. Anyway,

   volume = (22/7) * 6750

          = 21,214 ft^3

But in fact, when I use the 'pi' key on my calculator, I get the same
answer you did. So now I'm wondering: Why do you think your answer is 
wrong? One possibility is that you're supposed to convert the
answer into units other than cubic feet.  Is that the case? 

What else can you tell me about the problem?  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/06/2003 at 23:53:05
From: Dan
Subject: Cylinders

The problem was given just to find the cubic cm of the item.

Date: 02/07/2003 at 09:37:12
From: Doctor Ian
Subject: Re: Cylinders

Hi Dan,

Okay, thanks for getting back to me about this. There are lots of
possible ways to measure volume - gallons, liters, and so on.  But one
common way is to use cubic units, like cubic feet, or cubic inches. 

You found the number of cubic _feet_ in the cylinder. That is, suppose 
you had an empty cubic container, where each edge is 1 foot long. That 
would hold one cubic foot of water: 

  1 ft^3 = 1 ft * 1 ft * 1 ft

In order to fill the cylinder, you'd have to fill this cube and dump 
the contents into the cylinder a bunch of times, right? How many 
times? About 22,205 times. That's what you computed. 

Now, suppose I take away that cube, and give you a different cube,
where each edge is 1 cm long, instead of 1 foot long. The cube would
hold one cubic centimeter of water:

  1 cm^3 = 1 cm * 1 cm * 1 cm

In order to fill the cylinder, you'd have to fill this cube and dump 
the contents into the cylinder a bunch of times, right? How many 
times?  I think you can see that it's going to be a lot more than
22,205 times!

But how many more?  Well, let's think about how many of the 1 cm cubes
it would take to fill the 1 ft cube. We get the volume of a cube by
cubing the length of the side (i.e., raising it to the third power, as
I've been doing - this, by the way, is why raising something to the
third power is _called_ 'cubing').  

There are exactly 2.54 cm in an inch. (That's an exact conversion, and 
it's one of the few worth memorizing.) So if one edge of my cube is 1 
foot, it's also 12 inches, and therefore it's 12*2.54 centimeters 
long. Does that make sense? 

So my 1 ft cube is about 30.5 cm on a side. So the volume of the cube, 
in cubic centimeters, is 

  30.5 cm * 30.5 cm * 30.5 cm

which comes to about 28,373 cm^3.  

Now let's think of it this way. To fill the cylinder, you'd have to 
fill up the 1 ft cube 22,205 times. But to fill the 1 ft cube, you'd 
have to fill the 1 cm cube 28,373 times. So by the time you've filled 
the cylinder, you've filled the 1 cm cube

  22,205 * 28,373 = about 630 million

times. And this is what we mean when we say that the volume of the
cylinder is about 630 million cubic centimeters. 

Does that make sense? 

So you actually computed the correct volume; but you didn't use the
units required by the problem. And as you can see, units make a big
difference. Suppose you stopped to get gas, and you were asked how
much you wanted. You might say, '10'. But what have you asked for? 
10 gallons? 10 ounces? 10 liters? In a case where you're talking about 
length, or area, or volume, or any real world quantity, a number means 
nothing unless it's accompanied by some units - and the units we 
choose will determine the number we use.  

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/07/2003 at 10:36:38
From: Dan
Subject: Thank you (Cylinders)

Thanks so much for your time on this.  I am spreading the word about 
the site because you helped me a lot. I look smart to my 7th grader.