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currently a local newspaper company sells print subscriptions for $9 30 cents a month and has 2400 subscribers based on a survey conducted they expect to lose 20 subscribers for each 10 cent increase from the current monthly subscription price all right 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 but 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 it's going to be equal to the number of subscribers they have so as for subscribers times the price per subscriber part times the price per subscriber so that's going to be their income now in this little in the 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 10 cent increase from the current monthly subscription price and the current monthly subscription price is nine dollars 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 something to subtract 20 subscribers for every for every dime above 9 dollars and 30 cents so let's multiply that so if we just take our if we take P minus 9 dollars and 30 cents this would give us the absolute price increase how much we've gone above nine dollars and 30 cents and if we want to figure out how many dimes we are above nine dollars and thirty cents we could just divide that by ten cents 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 this is probably the most be 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 R is going to be equal to 2400 and then let's see 20 divided by 0.1 it's going to be 200 - 200 times P minus 9 dollars and 30 cents and then we can distribute this 200 so subscribers it's going to be equal to 2400 minus 200 P minus 200 P and the negative 200 times negative 9.30 let's see that's going to be 2 times 9 dollars 30 is $18 60 but it's going to be $1,800 and instead of $18 60 cents it's going to be 18 hundred and sixty dollars so plus 18 60 that's negative 200 times negative nine point three 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 see the 2400 plus 1860 if you had a thousand you get two 3400 and then you add 860 you get 240 260 so you get four thousand two hundred and sixty minus 200p 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 so this writing s we could write we could write 42 60 minus 200p and then times P times P now we can distribute we can distribute to this P and we're gonna get income as a function of price is four thousand two hundred and sixty P minus 200 P squared so income as a function of price it is a quadratic and I actually like - right my highest degree terms first so I'll just income is equal to I'm just gonna swap these negative 200 P squared plus forty two forty to sixty 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 if this is if this I mean so if this is the I axis that's the I axis if this is the P axis right over here we know that it's going to be so this is going to be the P axis we know it's going to 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 it was up or opening you'd be able to find a minimum point but then there's no it wouldn't be bounded on on to the upside 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 the P coordinate of the vertex of this parabola now let's just remind ourselves how we can find the vertex of a problem there's multiple ways to do it you can find the roots you could find the x-values that make 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 a if I have something of the form AI 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 it's going to be over 2a over two times negative two hundred so that's going to be negative four hundred and so what is that going to be getting 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 fortieths or we could write this as this is going to be equal to 10 and 13 13 13 twentieths 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 is this would be the same thing as 10 n let's see if you multiply the denominator by 5 you multiply the numerator by 5 10 and 65 hundredths 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 figure out well what are the what are the P values that 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 you could actually tackle that and you could do that just by looking at this original one you could say well when does this thing equal zero so you could say when does 40 to 60 P minus 200 P squared equals zero and since you have no constant term we actually could just factor out a P so you say P times 40 - 60 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 zero income if you if you don't charge anything or if 40 to 60 minus P is equal to oh let me be careful here I don't wanna forty to sixty 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 40 to 60 40 to 60 minus 200 P is equal to zero this is when you don't charge anything you're obviously gonna have no income and this is when you charge so much you're gonna lose all your subscribers and you'll have no income but let's see if you want to solve for if you add 200 P to both sides you get 200 P is equal to 40 to 60 divide both sides by 200 you get P is equal to 40 to 60 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 40 to 60 over 400 which is exactly what we figured out before that's $10.65