# Why use exponential equation in a (simple) rate problem?

I have a problem:

In the beginning there were 4. When time equaled 5, there were 20. How many would there be when time equaled 40?

To begin with, I really don't like this problem for its lack of units, making it very ambiguous. Also, what does this have to do with the exponential equation? It seems like a simple rate problem with an increase of $20-4$ every $5$ units of time.

By solving it like a rate problem I got the answer: $132$

But, by using the exponential equation my math textbook comes up with: $1,569,542$

$A_t = A_0e^{kt}$

$20 = 4e^{k5}$

$5 = e^{5k}$

$1.61 = 5k$

$0.322 = k$

$A_{40} = 4e^{0.322 \left(40\right)}$

$A_{40} = 1,569,542$

I am very confused as to the meaning of the problem.

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## 2 Answers

You have 2 points and you are trying to fit an equation to those two points. The answer you get depends on the equation you try and fit to those points. You've chosen a line, your book has chosen an exponential. Without a reason why you would prefer one equation over the other you and your book are both equally right.

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Thanks for the help. I would say my book is wrong for assuming an exponential growth. –  Web_Designer Mar 19 '13 at 23:35

Your reasoning is correct; you have to assume an exponential growth and calculations follow.

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