 This video will talk about rational applications. Rational functions are needed when we have a concept that involves an idea like per, since they involve fractions. An example would be speed, since it's miles per hour. Here's a few more. Some students agree to share equally to rent a large beach house for one week over spring break. The house costs $1,800 for the week, and if 15 students want to rent the house, how much is the cost per student? Well, they're going to divide it up evenly, because they said equally. So we have to take our $1,800 that we had and divide that by the number of people. And that the way it works, cost divided by the number of people will then give you that each person would have to pay $120. The PIVN represents the cost per student in dollars if in students rent the house. Well that's exactly what we just talked about. We have our $1,800 that the cost is divided by that number of students. So if we want to know, is it an increasing or is it a decreasing function? Well, if you think about it intuitively, as you get more and more people dividing up the amount of money, it should be a decreasing type of function. But let's just look at the graph and see what happens with it. $1,800 divided by X, and that must mean that I need to make my window go to say 2,000. And if I look at my graph then I can see that it is definitely decreasing. So we were right. So it's a decreasing function because the more people you get, the less money it's going to cost you. Finding P of 10, remember our function was 1,800 divided by N, but the N is now my 10. And if I just cancel off the zeros to divide by 10, then I just end up with each person has to pay $180. And exactly what this means in the situation, that's exactly it. Each person pays $180. Find N if P of N is equal to 90. So this time we know 90 and we have to solve for N. Well, remember to solve rational equations we have to multiply by the denominator. We only have one denominator here, so we only have to multiply by N. My N's cancel on this side, and this gives me 90 N equal to 1,800. And divide by the 90, we find out that N is going to be equal to 20. Which means 20 people splitting the cost would each pay $90. So we have this one. Suppose you're throwing a party and you want to share the expenses equally. So again, we're sharing expenses equally. Among all who attend, the band you hire costs $200 and everyone who comes will drink at least $3 worth of food. What does a total cost T for two people to attend? Well, the total cost is $200 for the band, plus the two people that have to show up. Times that $3 that they're going to eat and drink, or $206. Happens if five people show up while you still have that $200 for the band. But this time you have five people that you have to multiply by that $3. So you'll have $215. And if 20 people show up, again you have the flat fee of the cost of the band. Plus you have 20 people that are going to eat $3 worth of food. So now we have $260. How much should each person charge if two people attend? Go back and look. It was $206 for two people, dividing by two, and we have $103 each. That's a lot of money for a party. If we have five, that was $215 that we had to divide up now by five people. And that would go in 43 times, so we have $43 each, getting a little better. And when we had $260 divided by 20, it's $13 each. So what is a total cost function? Well, we've been doing that over and over again, so let's think of what we did. The cost is $200 for the band plus $3 of food times a number of people, right? So plus 3n. How much should be charged? The charge is going to be the $200 plus the 3 times the number of people. But now we're going to divide that evenly amongst those in people, those same in people. So finally, if you want to charge each person $5, so that's the charge, how many people need to attend? So we have 200 plus 3n divided by n. Again, we have to clear our fraction, so 5n is going to be equal to 200 plus 3n. And if I combine my ns by subtracting this 3n over here, then I'm going to have 2n is equal to 200, and n will be equal to 100. And remember, n is number of people, so 100 people would pay $5 each.