Population Question Using Half-Life Equation

[justine411]

Homework Statement

Vancouver Island has an area of 33000km^2 but no wombats. If we release a pair of wombats on Vancouver Island and their annual rate of increase is 50%, how many years will it be before the wombat population reaches one pair per km^2?

Homework Equations

[A]t=[A]oe^-kt t1/2=ln2/k

The Attempt at a Solution

This is in a rate of reaction tutorial package and the only equation I could think to use is the half life one, but have it going up by 1/2 instead of going down by 1/2. I'm not sure how to manipulate the equation though. The answer is 26 years, I'm just not sure how to do that. I'm not sure what to put in for k either. I've tried using 0.5 but that doesn't work. Any suggestions?

 

[Moonbear]

Okay, I can start off by pointing out this isn't a half-life problem. You're not talking about decreasing populations, you're talking about population growth, so while you're generally on the right track, an equation that calculates half life decay rates isn't going to be adequate. Do you know how that equation was derived so you can adjust it for population growth instead of decline?

The other thing you need to factor into this is that you have to first calculate your population size endpoint to solve for time. You are given all the information you need to calculate this based on the area of the habitat and the desired density of the population. So start there.

Although your question is based on a biological system, you don't need biology knowledge to solve this one, you need to know the math. I'd like to move it over to one of the math homework help sections, but will need your help to make sure I put it in the right one. Have you taken calculus yet (or are you in calculus now)? The equations you need to use are based on calculus derivations, but if you do not yet know what a derivative is, I will put it in the pre-calculus section so you get help based on your level of math knowledge.

 

[justine411]

I understand calculus, at least derivatives and integrals, so you can move it there if you want. Thanks SO much for your help. I thought it belonged here because it was in a chemistry tutorial, though it doesn't seem much like chem.

 

[Moonbear]

Okay, I'll make the move to the calculus HW forum. I think the folks there will be able to help you better than I can with this.

 

[Doc Al]

Homework Statement

Vancouver Island has an area of 33000km^2 but no wombats. If we release a pair of wombats on Vancouver Island and their annual rate of increase is 50%, how many years will it be before the wombat population reaches one pair per km^2?

You don't really need calculus, just a calculator (and a bit of trial and error).

First, find the pattern.

At n = 0 years, P = 1 (P = number of pairs)
At n = 1 years, P = (1.5)
At n = 2 years, P = (1.5)^2
...

Got it?