- Solving equations by graphing
- Solving equations by graphing: intro
- Solving equations graphically: intro
- Solving equations by graphing: graphing calculator
- Solving equations graphically: graphing calculator
- Solving equations by graphing: word problems
- Solving equations graphically: word problems
We can approximate the solutions of any equation by graphing both sides of the equation and looking for intersection point. See how we apply this idea to solve some word problems.
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- These two function at the end would never equal zero, would they ?(6 votes)
- If you are referring to the two populations 24exp(0.4t) and 9exp(0.6t) then you are correct, they are not zero at t = 0. At t = 0, the first one is 24, the second one is 9. Thinking about it physically, you can't grow a population until you have a non zero starter.(4 votes)
- Where does Sal get "e"? He's only talking about "f" and "g," so what does "e" do?(1 vote)
- e is a constant that is approximately equal to 2.71. Here, e is raised to the (0.4t) power, which will either increase the value of f(t) as t increases. For example, at f(1), the function equals 24 x e^(0.4), but at f(0), the function equals 24 x e^(0), or 24.(1 vote)
- [Instructor] We're told to study the growth of bacteria, a scientist measure the area in square millimeters occupied by a sample population. The growth of the population can be modeled by f of t is equal to 24 times e to the 0.4 times t power where t is the number of hours since the experiment began. Here is the graph of f. So I guess f is going to be, the output of this function is 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 going, and it clearly is growing, or it looks like it's growing exponentially. In fact, we know it's growing exponential 'cause 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? Well, 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, and now, we're told the same thing where you're using square millimeters, 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 equals 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. So and 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, they're talking about this point right over there. So does that describe the time when the population occupies 600 square millimeters? So that is the time when the population has, indeed, reached 600 square millimeters, 'cause that's the line y is equal to 600 square millimeters. So I like that choice, I will select it. The next choice. It gives the solution to the equation 24 times e to the 0.4 t is equal to 600. Well, if you think about it, this right over here in blue, we've already talked about it, that is y is equal to 24 times e to the 0.4 t power. This is y is equal to 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, it means that that is equal to that or that 24 times e to the 0.4 t power 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 of the population occupies is equal to the number of hours. That's definitely not the case, 'cause the area here is 600 square millimeters. 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 even 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, and it says, it says, so once again, we measure the area in square millimeters to figure out the growth of the population. The growth of, oh, it's here. We have two populations here. It says the growth of population A can be modeled by f of t is equal to that. We've seen that already, but now they are entering another population. The growth of population B can be modeled by g of t is equal to this, where t is the number of hours since the experiment began. Here, 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 they intersect right around there. So that's the point at which 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? And it looks like it happens after about five hours. So round to the nearest integer and say 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 it 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 this. 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 equals zero. When time t equals zero, 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 equals zero 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, the meaning of the intersection point of the graphs? But that doesn't tell us about what the starting situation was. It gives the solution to equation 24 times e to the 0.4 t is equal to nine times e to the 0.6 t. Well, we already talked about that in the last example where we only had one curve, and that actually is the case because y is equal to 24 times e to the 0.4 t is the curve for population A, and then y is equal to nine times e to the 0.6 t is the curve for population B, and so the point at which these two curves intersect, that's the point at which both this where the t value that gives the same, so that this expression will give you 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 the solution to the equation 24 times e to the 0.4 t is equal to zero? No, that would happen, if you want to know when it's equal to zero, you would look at the curve y equals zero. 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 to the 0.4 t, but that's not what this point of intersection represents, so we definitely wouldn't pick that one either.