00:01
Hi there, so for this problem, we are told that we start with an initial value that is the equals to zero that is equal to 15 and the population of a brady dog community grows at a rate that is given and that radius that the rate of change of the population with respect to time is equal to 20 minus the time divided by five and now for part a of this problem the question is what is the population 13 months later.
00:37
Okay.
00:39
So what we need to do first is to obtain the expression for the population.
00:44
Now since we are given the rate of change of the population with respect to time, which is this expression right here.
00:52
Now what we need to do is to separate the variables and thereby and integrate that expression.
00:59
So the population at any given time is the integral of 20 minus the time divided by 5 integrated over the time.
01:10
Now with this, we just need to simply solve this integral.
01:14
The solution of this integral is easy.
01:17
That will be just simply 20 times the time minus the time squared divided by 10, and this plus a constant of integration that we're going to call c.
01:29
Now, to determine the constant of integration, we evaluate this at 0, that we know that this will give us c, which we already know that this is equal to 50.
01:39
So with that, we will obtain that the population at any given time is equal to 20 times the time minus the time squared divided by 10...