# Exponential growth of populations (Q=Ae^kt)

1. Apr 28, 2007

### Alistair

1. The problem statement, all variables and given/known data
The number of rabbits in a colony is given by N=80e^(0.02t) where t is in days.
c) After how many days will there be 500 rabbits?

N=500
A=80
k=0.02
t=?

2. Relevant equations
(ln being the exponential logarithm)
Q=Ae^kt
and possibly the conversion formula: ln y = x --> y=e^x

3. The attempt at a solution

what i tried was coverting Q=Ae^kt to A ln Q=kt
Then divided both sudes by x to give:

A ln Q = t
k

Which after substitution looked like this:

80 ln 500 = t
0.02

Which gave t=24,858 days.
where as the answer is 92 days... :grumpy:

Last edited: Apr 29, 2007
2. Apr 28, 2007

### marcusl

This equation is wrong. Perhaps you are getting confused with your variables; there's no reason to introduce x, stick with A, k and t and try again.

3. Apr 29, 2007

### Alistair

Yeah i ment k not x.
in my maths book in the examples it has x. but it still doesnt work with k in there...

4. Apr 29, 2007

### danago

Start by dividing both sides by 80, so you isolate the exponent part. From there it might seem easier.

5. Apr 29, 2007

### Alistair

i dont know how dividing both sides by 80 will make it easier or alter the answer in any way...
i want to know if there is a problem with my working. im not sure if it is even the right formula...

6. Apr 29, 2007

### HallsofIvy

Staff Emeritus
If you really believe that using the letter "x" gives you a different equation that using the letter "k", you need to review basic algebra!

You have already been told that there is a problem with your "working"!

You are given the formula $N= 80e^{0.02t}$ so of course, that the correct formula. You are also told that N= 500 so the equation you want to solve is $80 e^{0.02t}= 500$. Surely, it would be an obvious first step to divide both sides by 80? After you have done that take the natural logarithm of both sides.

7. May 2, 2007

### Alistair

ok got it.
500 = 80e^0.02t
goes to
ln 6.25 = t
0.02

8. May 2, 2007

### HallsofIvy

Staff Emeritus