00:01
In this problem we are given the population model of a city p of t and this equals p of t equals 234, 252e raised to the power of 0 .15t where t is the number of years and we have to determine the number of years so that the population reaches 1 million.
00:20
So basically we have to determine t when p of t equals 1 million.
00:25
So i'm going to substitute p of t equals 1 million into this population model and determine the t.
00:37
So let's do that.
00:39
We place a p of t by 1 million.
00:41
1 million equals 1 followed by 6 zeros.
00:45
This is 1 million and this equals 234 comma 252 e raised to the power of 0 .15t.
00:55
Now to solve this equation for t, we can apply natural logarithful.
01:00
To both sides.
01:01
But before that, i'm going to divide this both sides by this quantity.
01:07
So let me do that.
01:08
So when i do that, i get 1 million divided by 234 ,252.
01:18
So this equals e raise to the power of 0 .15t.
01:22
Now we can apply the natural logarithm to both sides.
01:26
So i'm going to apply natural logarithm.
01:28
So write on this as natural logarithm.
01:30
In fact, i can also apply the natural logarithm to this side.
01:35
So therefore this becomes a natural logarithm of e raise to the power of 0 .15t.
01:40
Now we apply the power rule of natural logarithm.
01:44
So which means this power will come in front.
01:47
So in the next step, i can rewrite this as this quantity remains as it is...