Dimensional Analysis

Date: 10/06/2001 at 15:33:00
From: Katie
Subject: Dimensional Analysis

The question is...

Mr. R wants to give all of his chemistry students enough chocolate to 
make them goofy for the rest of the day. It's a known fact that it 
takes 47 chocolate chips to make a student goofy. The average cookie 
contains 29 chocolate chips. Mr. R has 5 classes and each class has 
an average of 26 students. How many toll house cookies does Mr. R 
need to make?

I have to use dimensional Analysis while doing figuring this question.  
How should I go about doing it?

Date: 10/08/2001 at 14:27:48
From: Doctor Greenie
Subject: Re: Dimensional Analysis

Hi, Katie -

Cool problem! I'm glad to see that dimensional analysis is being 
taught somewhere - it seems to be a topic that is unfortunately 
missing from the curriculum in many schools.  I have always found it a 
very useful tool, and this problem makes the topic fun.

Let's work this problem first without using dimensional analysis, so 
we will be able to confirm later that our solution to the problem 
using dimensional analysis is correct. (Or perhaps we will use our 
solution using dimensional analysis to confirm this solution...)

Mr R has 5 classes averaging 26 students; so he has 5*26 students. It 
takes 47 chocolate chips to make the average student goofy, so he 
needs 5*26*47 chocolate chips. Finally, the average cookie contains 29 
chocolate chips, so the number of cookies he needs is

    5 * 26 * 47
    -----------
        29

Now let's look at the problem using dimensional analysis.  We have the 
following information:

(1) number of classes = 5/1
(2) students/class = 26/1
(3) chips/student to make student goofy = 47/1
(4) chips/cookie = 29/1

I have written all these numbers as fractions because, with 
dimensional analysis, I may find I want to use, for example, 
classes/student = 1/26 instead of students/class = 26.

With dimensional analysis, I start by looking at what I am trying to 
determine. What I am after in this problem is the number of cookies 
required. In my given information, "cookies" appears only in (4): 
chips/cookie = 29/1. Since I want my answer to be "cookies," I need to 
use the reciprocal of the given information:

    cookies    1
    ------- = --
     chips    29

Now I see that I need to multiply this by something with "chips" in 
the numerator; (3) above tells me that chips/student = 47/1...

    cookies    chips    cookies    1   47
    ------- * ------- = ------- = -- * --
     chips    student   student   29    1

Now I see that I need to multiply this by something with "students" in 
the numerator; (2) above tells me that students/class = 26/1...

    cookies   students   cookies    1   47   26
    ------- * -------- = ------- = -- * -- * --
    student    class      class    29    1    1

I now have "cookies/class"; I need to multiply that by the number of 
classes to get "cookies" ...

    cookies                        1   47   26   5
    ------- * classes = cookies = -- * -- * -- * -
     class                        29    1    1   1

So, using dimensional analysis, we see that the number of cookies 
required is

     47 * 26 * 5
    --------------
          29

which agrees with the result we got above.

Thanks again for a problem that can be used nicely to demonstrate the 
use of dimensional analysis.

I hope this helps. Write back if you have any questions on this.

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