00:01
All right, a company finds that it sells since the company started in 2000 can be modeled by this function.
00:08
So let's write our function, s of t, is equal to 20 t squared plus 800 t plus 300, and that is all divided by 8t squared plus 10t plus 100.
00:33
So s is the total sales and t this is in millions and t is the number of years since 2000 years since 2000 so what that means is that for the year 2000 t is equal to zero now what were the cells in the year 2000 so again that means we're looking at s of zero and so so we're going to have, you know, this first term, 20t squared is going to go away because zero squared is zero, 20 times zero is zero, 800 times zero is zero.
01:21
So we have 300, and it's going to be the same thing in the bottom, and we get over 100.
01:27
That is equal to $3 million.
01:31
So remember, our money is in million.
01:33
So in the year 2000, we had $3 million in sales.
01:40
After how many years, what does the model predict, or after many years, what does the model predict cells will be? well, if you think about how this graphs, this is a rational function where the degree of our numerator is equal to the degree of our denominator.
01:59
So that means that we have a horizontal asymptote at, you take these first two leading coefficients.
02:09
So we get a horizontal asymptote at 20 over 8, which reduces to 10 over 2, which is 2 .5, if we write it as a decimal.
02:32
So what's going to happen, if we're talking about many years, x is our years, as x goes often to infinity, our function is going to approach this 2 .5.
02:43
So after many years, the model predict cells will be $2 .5 million.
02:52
Calculate the years when the cells are 9 million algebraically.
02:57
All right, so we just have to plug in 9 for this s of t.
03:04
Our cells are given in million, so 9 is equal to, and it's equal to this function right here.
03:13
Actually, i'm going to make things, i'm going to take, kind of skip a step.
03:17
I'm going to go ahead and multiply both sides by the denominator of my function, because i definitely want to get that out of the denominator.
03:27
And that's going to be equal to 20 t squared plus 800t plus 300.
03:35
Let's distribute this 9.
03:43
And so that's equal to the same thing, 20t squared plus 800t plus 30.
03:51
And then let's get everything over to one side so that we have our function equal to zero.
03:58
So 72 minus 20 is 52 t squared.
04:03
90 minus 800.
04:06
It's going to be minus 710 t.
04:12
And then 90, this would be a minus right here.
04:15
So it's going to be plus 870.
04:18
And that is equal to 0...