Solving equations by graphing: word problems | Algebra 2 | Khan Academy

We're told to study the growth of bacteria. A scientist measures the area in square millimeters occupied by a sample population. The growth of the population can be modeled by ( f(t) = 24 \times e^{0.4t} ) where ( t ) is the number of hours since the experiment began.

Here's the graph of ( f ). So I guess ( f ) is going to be the output of this function; it's going to be the number of square millimeters after ( t ) hours.

All right, so here we have the graph. We see how, as time goes on, the square millimeters of our little bacterial population keeps growing, and it clearly is growing—or it looks like it's growing—exponentially. In fact, we know it's exponential because it's an exponential function right over here.

And they say: when does the population first occupy an area of 400 square millimeters? So pause this video and try to figure that out.

All right, and this is a screenshot from the Khan Academy exercise, so we want to say when does the population first occupy an area of 400 square millimeters? Let's see, 400 square millimeters is right over there, and so it looks like after seven hours that we are going to be 400 square millimeters or larger. So it first hits it after seven hours—so seven hours, just like that.

Now, let's do the next few examples that build on this. So if I go back up to the top, we are told the same thing; we're using square millimeters to study the growth. This is the function, but then they add this next line: here is the graph; here is the graph of ( f ) and the graph of the line ( y = 600 ).

So they added that graph there and then they say: which statement represents the meaning of the intersection point of the graphs? All right, so let's look at the choices here. It says choose all that apply, so pause this video and see if you can answer that.

All right, so choice A says it describes the time when the population occupies 600 square millimeters. So which statement represents the meaning of the intersection point of the graph? So they're talking about this point right over there.

So does that describe the time when the population occupies 600 square millimeters? That is the time when the population has indeed reached 600 square millimeters because that's the line ( y = 600 ). So I like that choice; I will select it.

The next choice gives the solution to the equation ( 24 \times e^{0.4t} = 600 ). Well, if you think about it, this right over here in blue—we've already talked about it—that is ( y = 24 \times e^{0.4t} ). This is ( y = 600 ), so the ( t ) value at which these two graphs equal—that means that they're both equal to the same ( y ) value—or another way to think about it is that ( 24 \times e^{0.4t} ) is indeed equal to 600.

So I like this too; it gives a ( t ) value where this is true. So that's the solution to that equation. It describes the situation where the area the population occupies is equal to the number of hours. That's definitely not the case because the area here is 600 square millimeters, and the hours looks like it's a little bit after eight, so they're definitely not equal.

It describes the area the population occupies after 600 hours? No, we don't have to look up there; this ( t )-axis doesn't even go to 600 hours, so we wouldn't select that as well.

Now let's keep building and go to the next part of this. It says once again we measure the area in square millimeters to figure out the growth of the population. The growth of—oh, so here we have two populations. Here it says the growth of population A can be modeled by ( f(t) = ...). We've seen that already, but now they are introducing another population.

The growth of population B can be modeled by ( g(t) = ... ) where ( t ) is the number of hours since the experiment began. Here are the graphs of ( f ) and ( g ). So now we have two populations; they're both growing exponentially but at different rates.

And then it says: when do the populations occupy the same area? It says round your answer to the nearest integer, and you could pause this video and try to think about that if you like. Well, you can see very clearly that it looks like—or they intersect right around there—so that's the point where they're going to occupy the same area. It looks like it's about 175 square millimeters, but they're not asking about the area; they're saying when does it happen?

It looks like it happens after about five hours, so round to the nearest integer, it's five hours.

Now let's do the last part. So it's the same setup, but now they are asking us a different question. They are asking us which statements represent the meaning of the intersection points of the graphs. All right, so choice A says—and then pause the video again and try to answer these on your own.

All right, choice A says it means that the populations both occupied about 180 square millimeters at the same time. So let's see; that looks about right. I had estimated 175, but we could call that 180, and it looks like that does roughly happen around the fifth hour.

So it looks like they're occupying the same area at around the same time, so I like that choice. It means that at the beginning, population A was larger than population B. Well, the point of intersection doesn't tell us what population was larger to begin with.

We could try to answer it by looking over here. When time ( t = 0 ), population A is the blue curve—it is ( f )—and so it does look like population A was larger than population B at time ( t = 0 ), at the beginning.

But that's not what the point of intersection tells us. So they're not just asking us for true statements; they're saying which statements represent the meaning of the intersection point of the graphs. But that doesn't tell us about what the starting situation was.

It gives a solution to the equation ( 24 \times e^{0.4t} = 9 \times e^{0.6t} ). Well, we already talked about that in the last example where we only had one curve, and that actually is the case because ( y = 24 \times e^{0.4t} ) is the curve for population A and then ( y = 9 \times e^{0.6t} ) is the curve for population B.

So the point at which these two curves intersect—that's the point at which both—the ( t ) value that gives the same ( y ) value as this expression. Or another way to say it is we're at the ( t ) value where this is equal to this. So it does indeed give the solution to the equation.

And then the last choice is: it gives a solution to the equation ( 24 \times e^{0.4t} = 0 ). No, that would happen if you want to know when it's equal to 0; you would look at the curve ( y = 0 ), I'll do that in a different color, which is right over here, and see where it intersects the function ( f ), which is equal to ( 24 \times e^{0.4t} ).

But that's not what this point of intersection represents, so we definitely wouldn't pick that one either.