Electrical Relationships

Date: 12/10/96 at 17:30:06
From: MR MICHAEL   FISHMAN
Subject: Algebra 2 / Advanced Algebra

Hi,

I am currently taking Algebra 2 / Advanced Algebra.  I have tried many
different procedures to solve this problem but have failed each time. 
Here it is:

The resistance R in an electrical circuit is related to the diameter d
of the wire and the length L of the wire.  In an experiment, John
obtained the following data with a 50 ft. wire:

Diameter (inches)   .05  .08  .11  .14  .17  .2
Resistance (ohms)  9.0  3.5  1.9  1.1  0.8  0.6

How does R vary with d?

I believe the answer is supposed to be jointly, directly, or 
indirectly, or something along those lines. However, since no formula
was provided, I need to determine that first before I can tell if it's
direct, indirect, joint, etc. 

I have tried numerous things. I know that whatever the formula is, it 
will equal R. In this case, I chose .05 and 9. I tried .05*50, 50/.05, 
.05/50. I cannot seem to come out with 9. I also tried the same 
formulas with .08 and 50, but failed to get 3.5. 

Can you help me figure this out?  Thanks a lot for the help, and keep 
up the good work!

Mike

Date: 12/11/96 at 20:58:31
From: Doctor Rob
Subject: Re: Algebra 2 / Advanced Algebra

Well, first of all, it is clear that R varies inversely with d, 
because when d gets bigger, R gets smaller. For a fixed value of L, R 
will equal some function of d which gets smaller as d gets larger.  
This will eliminate things like a positive constant times d, which 
must vary directly with d, not inversely. You can try C/d, where C is 
a constant. Picking any of the above points will allow you to compute 
C at that point. Then you can test by checking other points. If this 
doesn't check, try a constant over some other function of d. Note 
that the value of L doesn't matter for this problem since it is the 
same at every point used in the experiment.

Another way to proceed is the following:

If you suspect that R = C*d^n for some n, you can take logarithms of 
both sides of this equation to get:

  log R = log C + n*log d

By plotting points (log d, log R) on a graph, you should be able to
see a straight line with slope n and log C as its (log R)-intercept.  
Once you get n and C, you can test the points from the experiment to 
see if they are close to the values for this function.  If so, then 
the formula R = C*d^n is a correct guess.  If not, then the function 
you seek must have another form, perhaps C*e^(-n*d), or C/log d, or 
something else.

If you need more assistance, please write back.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/