Exponential growth of populations (Q=Ae^kt)

[Alistair]

Homework Statement

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=?

Homework Equations

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

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...

 

[marcusl]

Homework Statement

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=?

Homework Equations

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

The Attempt at a Solution

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

A ln Q = t
x

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.

 

[Alistair]

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.

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

 

[danago]

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

 

[Alistair]

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

i don't 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. I am not sure if it is even the right formula...

 

[HallsofIvy]

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

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!

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

i don't 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. I am not sure if it is even the right formula...

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

You are given the formula [itex]N= 80e^{0.02t}[/itex] so of course, that the correct formula. You are also told that N= 500 so the equation you want to solve is [itex]80 e^{0.02t}= 500[/itex]. 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.

 

[Alistair]

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

 

[HallsofIvy]

Okay, now, what is your answer to the question?