Expert: Scott A Wilson - 11/5/2009

Question
QUESTION: Suppose that the number of cars, C, on 1st Avenue in a city over a period of time t, in months, is graphed on a rectangular coordinate system where time is on the horizontal axis. Suppose that the number of cars driven on 1st Avenue can be modeled by an exponential function, C= p * at (C=p*a^t) where p is the number of cars on the road on the first day recorded. If you commuted to work each day along 1st Avenue, would you prefer that the value of "a" be between 0 and 1 or larger than 1?  I don't even know where to start.  Thank you advance for your help.

ANSWER: Since the equation we are given is C = p•a^t, the value of a needs to be less than one for the function to return smaller values as time goes on.

Using a spreadsheet and exporting the data to you,
I will show you the effect if a = 0.95 and a=1.05.

This is a^t where t is 1, 2, 3, 4, 5, ... 98, 99, 100.

As can be seen, 0.95 decreases where 1.05 increases.

You might think of this as being one means you get the whole thing again.
Being slightly greater than 1 means giving you a little more than 1.
Being less than 1 means we're taking away just a little and not quite giving back the whole.


---------- FOLLOW-UP ----------

QUESTION: Could you check this.  I was trying to work on this while waiting for your response.  I think I did it right not sure though.  This is what I came up with (it is funny I was just finishing up when I saw your email)

The first equation will represent “a” being between 0 and 1(we will use 1) where “p” =15 (number of cars on the road) and “t” is time (in months) which will be 5.
C=p∙a^t
C=15∙1^5
C=15∙1
C=15
The second equation will represent “a” being larger than 1(which we will use 5) where “p” = 15 (number of cars on the road) and “t” is time (in months) which will be 5.
C=p∙a^t
C=15∙5^5
C=15∙3125
C=46875

Thank you a million for your help.  I rate you at 210% plus.  

Answer
I can tell this is going to say , 'You're welcome.'  Yet if you want to read on, there's more stuff that is foaming out of my brain...

Problem
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Yeah, that looks right.  Basically it all comes down to the fact that the critical value is at 1.

Percent
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Now it is not usually possible to get over 100% correct.

With the teachers giving extra credit problems on their tests,
though, they say the regular test all right is 100% plus anything you do on the extra problem.

This allows for a score of over 100.

Anywaya, if I take the 210%, break it in two, I can give you back 105%,
so we're both got the test and extra credit right.  No?

Powers of 5
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By the way (off on another tangent) I recognize 46,875 as being 3*5^6.
For some reason I don't know, I know the powers of 5 as 5, *5=25, *5=125, *5=625, *5=3,125,
*5=15,625, *5=78,125, *5 = 390,625, etc.

Powers of 2
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Another one is the powers of 2.  See, 2^10 = 1,024, so a kilobyte is only approximately a 1000 bytes.  It is really 1,024 bytes.  A megabyte, in the same way, is said to be 1 million bytes.  That is because 2^20 = 1,048,576.  Yeah, you probably think I'm pretty wierd, but that's the kind of stuff I know.