Exponential word problem (i think)

[Jacobpm64]

I'm not sure if this is calculus, but it is like a review in my calculus class.

Suppose that in any given year, the population of a certain endangered species is reduced by 25%. If the population is now 7500, in how many years will the population be 4000?

I've known how to do this before. I just forgot how to set it up. I'm pretty sure it's with exponents though. Please help.

 

[LeonhardEuler]

If it appeared as a question for calculus review, then they probably want you to proceed like this: You know the population can be descrided by an exponential equation of the form
P(t)=Ce^{kt}
for unknown constants C and k. Knowing the initial population will allow you to solve for C. Knowing the population a year later will let you solve for k. Then solve for time in the equation by taking ln's. Plug in the population you want and get the time it takes.

 

[Jacobpm64]

so let's see..
P(t) = 4000
C = 7500
e = the constant 2.718 etc.
k = what's k? I'm not sure how to do it since it's decay...?
t = what I'm solving for

so what is k?

 

[LeonhardEuler]

P(t) = 4000

Not exactly:
P(t)=7500e^{kt}
And in one year the population will be reduced by 25%, so:
P(1)=7500-.25*7500=5625=7500e^{k*(1)}
So now you can solve for k.

 

[Tide]

It might be easier to start with

P = P_0 \left( \frac {3}{4} \right)^n

where n is the number if years.

 

[LeonhardEuler]

It might be easier to start with
P = P_0 \left( \frac {3}{4} \right)^n
where n is the number if years.

Yeah, I was thinking that too, but when I did that in high school calc class I remember my teacher took off points because she said "This is calculus class, solve the problem with calculus". They are probably trying to teach exponentials with this problem, but its good to know how to do it both ways.

 

[Jacobpm64]

2.1850811 years?

I believe so.. confirmation?

and thanks a lot :)

 

[LeonhardEuler]

2.1850811 years?
I believe so.. confirmation?
and thanks a lot :)

That's right.

 

[HallsofIvy]

This was also posted under "homework".


1) Please do not post the same thing in two different places.

2) There is a "sticky" thread at the top of this area that says this is NOT the right place to post homework problems.

 

[Jacobpm64]

yeah, sorry about that, it won't happen again.. i noticed that sticky after i posted this one.. so i reposted in homework.. all of my future ones will go in homework.. thanks for the help, nonetheless