00:01
A wood rat population rose from 200 to 230 in one year.
00:04
Assume that the population growth rate continues in this geometric or exponential way.
00:10
Find the population in 10 years after the time when the population was 200.
00:15
So first we write the equation template.
00:19
The number or population as a function of time would be the initial value a times e to the exponents of r t.
00:34
Where r can be the exponential growth rate.
00:38
So we know that n of 0 where we have a times e to the r times 0 where t is 0.
00:52
We know that that's the initial value.
00:54
We know that that's given to be 200 at time t equals 0.
00:59
So that would give us a times e.
01:04
R times 0 is just 0 and that's equal to 200.
01:07
And e to the 0 power is 1 so that gives us a to be equal to 200.
01:13
So we know that initial value is 200.
01:15
The a value and then we can find the r value by using the fact that when t equals 1 the number or the population is 230.
01:27
So let's use that.
01:29
We can say that the n, the number at 1 would be the a we know is 200.
01:35
So 200 times e to the r times t is 1.
01:40
We know that that's equal to 230.
01:42
So now we can solve for r.
01:45
So that gives us 200 times e to the r equals 230.
01:50
And then we can divide both sides by 200.
01:52
That gives us e to the r equals 230 divided by 200.
01:56
And then we can rewrite this as e to the r equals 23 over 20.
02:04
And if we rewrite this by taking the log of both sides that brings this r down and that gives us r times the natural log of e to be the natural log of 23 over 20.
02:17
And then the natural log of e is just 1.
02:21
So that gives us the r value.
02:23
R is the natural log of 23 over 20...