Quadratic and exponential word problems — Harder example

- [Instructor] Currently, a local newspaper company sells print subscriptions for $9.30 a month and has 2400 subscribers. Based on a survey conducted, they expected to lose 20 subscribers for each $0.10 increase from the current monthly subscription price. That's interesting. What should the newspaper company charge for a monthly subscription in order to maximize the income, in order to maximize the income from the print newspaper subscriptions? So let's think a little bit about their income. It's actually their revenue, how much money they're bringing in, but I get what they're talking about. So their income, let's just say I for income, is going to be equal to the number of subscribers they have, so S for subscribers, times the price per subscriber, times the price per subscriber. So that's going to be their income. Now in this little, in the problem, they tell us that the subscribers themselves are going to be a function of price, and that makes sense. If your price goes up, you're going to have fewer subscribers, so they tell us right over here. Based on a survey conducted, they expect to lose 20 subscribers for each $0.10 increase from the current monthly subscription price. And the current monthly subscription price is $9.30. So let's see if we can write our number of subscribers as a function of price. So at the current price, we have 2400 subscribers. We have 2400 subscribers. But we're going to lose 20 subscribers, so lose, so I'm gonna subtract 20 subscribers for every, for every dime above $9.30. So let's multiply that, so if we just take our, if we take P minus $9.30, this would give us the absolute price increase, how much we've gone above $9.30, and if we want to figure out how many dimes we are above $9.30, we can just divide that by $0.10. So this part right over here, this tells us how many dimes above our current price is P, and then for every one of those dimes, we're going to lose 20 subscribers. So this is probably the most interesting part of this problem. This is kind of the crux of it. How do you set up, how do you set up subscribers as a function of price? Because once you do this, you can then substitute back in here and then you're going to have income purely as a function of price. Let me show you what I'm talking about. So before I actually even do that, let me simplify this a little bit. So subscribers is going to be equal to 2400, and then, let's see, 20 divided by 0.1 is going to be 200 minus 200 times P minus $9.30. And then we can distribute this 200. So subscribers is going to be equal to 2400 minus 200 P, minus 200 P, and then negative 200 times negative $9.30, let's see, that's going to be 2 times $9.30, that's $18.60, but it's going to be $1800 and, instead of $18.60, it's going to be $1860. So plus 1860, that's negative 200 times negative 9.3. And now we can add, we can add this to that, and we are going to get, we're going to get the subscribers are going to be equal to 2400 plus 1860. If you had 1000, you'd get to 3400 and then you get 860, you get 240, 260. So you get 4,260 minus 200 P. So we now have a simplified subscribers as a function of price, and now we can substitute this expression in for S, in for S in our original equation. So if we do that, we get income is equal to, instead of writing S, we can write, we can write 4260 minus 200 P and then times P, times P. Now we can distribute, we can distribute this P and we're gonna get income as a function of price, is 4260 P minus 200 P squared. So income is a function of price, it is a quadratic, and I actually like to write my highest degree terms first, so I'll just, income is equal to, I'm just gonna swap these, negative 200 P squared plus 42, 4260 P. So this is a quadratic, and it is a downward opening quadratic. We know that because the coefficient on the second degree term is negative. So this graph, I as a function of P, so this is, if this, so if this is the I axis, that's the I axis, this is the P axis right over here, we know that it's gonna be, so this is gonna be the P axis, we know it's gonna be a downward opening parabola and that's good because we want to find a maximum point. A downward opening parabola will actually have a maximum point. If it was upward opening, you'd be able to find a minimum point, but then there's no, it wouldn't be bounded onto the up side. So we need to figure out, we need to figure out what price gets us to this maximum point, and this price, this is going to be the vertex, this is going to be the P coordinate of the vertex of this parabola. And let's just remind ourselves how we can find the vertex of a parabola. There's multiple ways to do it. You can find the roots. You can find the x-values that make this function equal zero and then the vertex is going to be halfway between those points. That's one way to do it. Another way to do it is you could say, look, if I have something of the form I is equal to a P squared plus b P plus C, instead of writing y and x, I wrote I and P, well, the vertex is going to be the coefficient, the negative of this coefficient, negative b over two times a. That's going to be the P coordinate of the vertex. P equals that is going to be when you hit the minimum or maximum point. In this case, it's going to be the maximum point. So what's negative b? B is this right over here, so negative b is 4260. That's gonna be over 2 a, over 2 times negative 200, so that's gonna be negative 400. And so, what is that going to be? Negative divided by negative is a positive. This is going to be the same thing as 426 divided by 40. And let's see, can I simplify this? Well, let me just see, this is going to be 40 goes into 426 10 times with 26 left over. So it's going to be 10 and 26/40, or we can write this as, this is going to be equal to 10 and 13, 13, 13/20. And so, if we wanted to write that in terms of dollars because we're talking about a price, we would want to write this in terms of hundredths, so this would be the same thing as 10 and, let's see, if you multiply the denominator by five, you multiply the numerator by five, 10 and 65/100, which is the same thing as $10.65. And luckily, we see that choice, we see that choice right over there. Another way we could have done it is we could have figured out, what are the P values that it gets to zero income, and then the one that's halfway in between those two is going to be where we hit our maximum. Halfway between your two roots are when you hit the maximum point, so that's another way that you could actually tackle that and you could do that just by looking at this original one. You can say, when does this thing equal zero? So you can say, when does 4260 P minus 200 P squared equal zero? And since you have no constant term, you actually can just factor a P, so you say, P times 4260 minus P is equal to zero. So this is going to be equal to zero either when P equals zero, and that makes sense. You're gonna have zero income if you don't charge anything. Or if 4260 minus P is equal to. Oh, I'm going to be careful here. 4260 minus 200 P, minus 200 P is equal to zero. So that's going to be equal to zero either when P is equal to zero or, or when 4260, 4260 minus 200 P is equal to zero. This is when you don't charge anything, you're obviously going to have no income. And this is when you charge so much, you're gonna lose all your subscribers and you'll have no income. Let's see, if you want to solve for, if you add 200 P to both sides, you get 200 P is equal to 4260, divide both sides by 200, you get P is equal to 4260 over 200. That's this price over here. That's the price at which you lose all your subscribers. So your maximum point is going to be halfway in between these two, halfway between zero and this, halfway between zero and that. Well, halfway between zero and that is just going to be half of this. So your maximum point is going to be at 4260 over 400, which is exactly what we figured out before. That's $10.65.