00:01
In this problem, we're given that a university is trying to decide a price for their football tickets, and we're given the following information.
00:10
At a ticket price of $20, they have an average of 40 ,000 people per game, and if every decrease by $2 of ticket price adds 10 ,000 people to the attendance, and each person spends on average $5 on concessions.
00:27
So we want to decide what price would maximize our revenue, and then how many people would be attending the game at that price.
00:34
So we need a revenue function, and so before we get to that, we probably want to write our demand function for the price.
00:40
So our price is going to equal every decrease by $2, so we're going to assume that it's linear, and we know that we want our y to be the price, and x is going to be, well, p is, i guess p equals y, and x is our number of people.
01:09
So when i make my slope, i want to make sure that it's the change in price over the change in people.
01:13
So when i decrease by $2, i'm adding 10 ,000 people.
01:19
So our slope is going to be negative 2 over 10 ,000.
01:23
Oops, that should be an x, and then we need to figure out what our y -intercept is.
01:27
They give us a point that's on the line, so i'm going to use that to find what i should put in for b.
01:34
So $20 will give me, i'm going to go ahead and reduce this to 1 over 5 ,000 times 40 ,000 plus b.
01:49
So 20 is going to equal 40 ,000 divided by 5 ,000 is 8, so negative 8 plus b.
02:00
We'll add our 8 over, so b is going to be 28.
02:04
So our demand function is going to be negative 1 over 5 ,000 x plus 28.
02:16
So that's going to give us the price of our tickets.
02:19
And so our revenue function here is going to be the price of our tickets times the number of people who show up plus 5 times that same number of people that show up, because each person spends $5 on concessions.
02:37
So if i, i'm going to go ahead and erase this for now, if i plug in my p of x, that gives me negative 1 over 5 ,000, and then multiply by x would be x squared plus 28x plus 5x.
02:56
And then i can simplify that a little bit more by combining my like terms.
03:02
So that's going to be 33x.
03:07
And then i want to maximize my revenue, so i need to find my first derivative so i can find my critical points.
03:12
So this will give me negative 2 over 5 ,000 x plus 33.
03:20
And i want to set this equal to zero to find my critical points.
03:24
So i'm going to move my 33 to the other side, and i'm going to go ahead and reduce this fraction.
03:30
So that'd give me negative 1 over 2 ,500.
03:34
And we have our x still...