00:01
In this question, we are told that the population of a certain city was 24 ,000 people in 1920 and 31 ,000 people in 1930.
00:12
And we are asked to find the population of the city in 2000, assuming that the population grows exponentially with a constant rate.
00:21
So let's assume that 1920 corresponds to t equals zero.
00:26
Then 1930 corresponds to t equals 10, because it's 10 years.
00:31
After 1920 and t equals 2000, 2000 corresponds to t equals 80 because it's 80 years after 1920 now we'll rephrase everything in terms of the function p which represents the population so p of 0 equals to 24 ,000 p of 10 equals to 31 ,000 and we are asked to find p of 80 in this question now, since the population grows exponentially, its formula should be p of t equals to p knot multiplied by a to the power of t.
01:25
To find p not, we need to plug in t equals zero.
01:30
On the one hand, p of zero equals to 24 ,000.
01:41
On the other hand, if you plug in t equals zero in the previous formula, you are going to get p knot multiplied by a to the zero.
01:49
And since a to the 0 equals to 1, you are going to get that p0 equals to 24 ,000.
01:56
Therefore, we can rewrite the formula for p as 24 ,000 times a to the power of t.
02:05
Now we will use the second condition to find the constant a.
02:10
P of 10 equals to 31 ,000.
02:19
On the other hand, if you plug in t equals 10 in the formula, you are going to get 24 ,000 multiplied by a to the 10th power.
02:28
And this gives us an equation for finding a.
02:31
We are going to get that a to the 10th equals to 31 ,000 divided by 24 ,000, which simplifies to 31 over 24.
02:47
And therefore, we can, after taking the, applying the one tenth power, we are going to get that a equals to 31 over 24 to the 110th power...