00:01
Hi there, so for this problem we are told that a certain population follows an exponential model and that exponential model p is in the following way, the initial population and then this times a elevated to t.
00:16
So the times t is the time past 2000 and p is the population in millions.
00:24
Okay, so then okay now if the population in the year 2002, so that will be by the year 2 because that will correspond to 2002, the population was, so the population at 2 is 40, 40 million okay and in the year 2004 is 160.
01:05
Once we have this for part a of this problem the question is to find the values of the initial population in a.
01:13
Okay, so first of all we can use these two conditions okay, so if we evaluate the function that we are that we are given at these two conditions we obtain the following the initial population times a elevated first at 2, so we know that this is equal to 40 and the other condition is that when we evaluate this at 4 this is equal to 160.
01:41
Okay, those are the two conditions given so we have in here what we can do is for example we can this one we can divide it by the previous one so that will be 160 divided by 40 that will give us 4 and then we can cancel some a in here so that will give us a square.
02:13
So finally solving for a is equal to the square root of 4 which is equal to 2.
02:21
Okay, so we determine the value of a.
02:23
P sub 0 the population we just now evaluate one condition in here so for example let's use this so the initial population then will be 40 divided by a to the square so that will be 4 so the population then initial population was 10.
02:45
So with this we can we obtain that this is the initial population 10 millions and a equals to 2...