The following function represents the profit P(n), in dollars, that a concert promoter makes by selling tickets for n dollars each:
P(n) = -250n^2 + 2,500n - 5,250
Part A: What are the zeroes of the above function, and what do they represent? Show your work.

- anonymous

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- anonymous

@SolomonZelman

- SolomonZelman

What does n represent?

- anonymous

dollars

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- SolomonZelman

precisely, n, is a price per ticket.

- SolomonZelman

What does P(n) represent?

- anonymous

profit

- SolomonZelman

Yes, when P(n) is 0, that means that the profit is zero (i.e. made nothing), right?

- anonymous

yes

- SolomonZelman

The x-intercepts of this function are the values of
n (or the value of the price per ticket) with which
the P(n) (or the profit) is 0.
(There are going to be 2 x-intercepts)

- anonymous

okay

- SolomonZelman

So, if you find the x-intercepts, you are going to find the 2 possible values of a price per ticket with which the promoter will NOT profit (but will not lose).

- SolomonZelman

So, do you see what these x-intercepts would mean?

- anonymous

yes

- SolomonZelman

Ok, now you have to solve for these x-intercepts.

- anonymous

the x intercepts are 3 and 7?

- SolomonZelman

yes, very good!

- SolomonZelman

Perfect!

- SolomonZelman

So, when the ticket costs 3$ or 7$ , then the promoter doesn't gain or lose.

- anonymous

now what?

- SolomonZelman

when the *average* ticket
I should say

- SolomonZelman

Well, all you had to do is to find these zero (these x-intercepts), and to then say what these x-intercepts represent in your situtation.

- SolomonZelman

you found x=3, and x=7 ((( and that is correct )))
now you have to say what do these intercepts mean/do/represent?

- SolomonZelman

just say what we have agreed on in the beginning.

- anonymous

can you help me with part B?

- SolomonZelman

what is part B?

- SolomonZelman

I can try my best with it:)

- anonymous

Part B: Find the maximum profit by completing the square of the function P(n). Show the steps of your work.

- SolomonZelman

Oh, good

- SolomonZelman

```
SIDE - NOTE:
Most people would say that this question requires calculus, but in a case
of a parabola (a quadratic does not). (this what most people say is a mistake).
```
ALL YOU GOT TO DO:
You have to write your P(n) in a vertex form, and the vertex is going to be the maximum.

- anonymous

so (5,1000)?

- SolomonZelman

Let me see.... i wil be typing, and if it is right I will post the work, and if it is not, then I will ask you to redo it and follow along with me.

- SolomonZelman

hope that is fair.

- anonymous

yes

- SolomonZelman

\(\large\color{black}{ \displaystyle P(n) = -250{\rm n}^2 + 2,500{\rm n} - 5,250 }\)
\(\large\color{black}{ \displaystyle P(n) = -250({\rm n}^2 -10{\rm n}) - 5,250 }\)
\(\large\color{black}{ \displaystyle P(n) = -250({\rm n}^2 -10{\rm n}+25-25) - 5,250 }\)
\(\large\color{black}{ \displaystyle P(n) = -250({\rm n}^2 -10{\rm n}+25)+(-250)(-25) - 5,250 }\)
\(\large\color{black}{ \displaystyle P(n) = -250({\rm n}^2 -10{\rm n}+25)+1000 }\)
\(\large\color{black}{ \displaystyle P(n) = -250({\rm n}-5)^2+1000 }\)

- SolomonZelman

yes, that is correct

- SolomonZelman

And quite quick

- anonymous

1 graphed it

- SolomonZelman

oh, you graphed it... good, but it is very good (I would insist that even required) to know that without graphing....
in any case though, lets do a little review of what you have actually found just now,

- anonymous

the axis of symmetry would be 5 right? (Thats part C)

- anonymous

okay

- SolomonZelman

yes part C is correct

- SolomonZelman

now the review of what does the vertex of (5,1000) actually mean in this case.

- anonymous

thats the maximum about of profit the promoter can make?

- SolomonZelman

that is you are refering to the point to a 5 or a 1000 ?

- anonymous

1000 makes more sense 1 think

- SolomonZelman

yes

- SolomonZelman

and what does 5 mean, can you tell me?

- anonymous

the amount the tickets cost?

- SolomonZelman

yes

- SolomonZelman

So when the average ticket costs 5$, you get a maximum profit of 1000

- anonymous

makes sense

- SolomonZelman

A quite small, if not miserable profit in a real life situation... but won't be talking about a real life situation because such situations are never modeled with quadratics in real life....

- SolomonZelman

yeah, so you got any questions regarding any of the PARTS ?

- anonymous

nope thats it

- anonymous

as usual you're the most helpful person on the whole site. Thanks!

- SolomonZelman

\(\large{\bbox[5pt, cyan ,border:2px solid black ]{ \rm Thanks! }}\)

- SolomonZelman

I wouldn't say I am most helpful, but won't deny some use.; you are always welcome!

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