Question

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.

Answer 1

@SolomonZelman

Answer 2

What does n represent?

Answer 3

dollars

Answer 4

precisely, n, is a price per ticket.

Answer 5

What does P(n) represent?

Answer 6

profit

Answer 7

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

Answer 8

yes

Answer 9

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)

Answer 10

okay

Answer 11

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).

Answer 12

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

Answer 13

yes

Answer 14

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

Answer 15

the x intercepts are 3 and 7?

Answer 16

yes, very good!

Answer 17

Perfect!

Answer 18

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

Answer 19

now what?

Answer 20

when the *average* ticket

I should say

Answer 21

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.

Answer 22

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

Answer 23

just say what we have agreed on in the beginning.

Answer 24

can you help me with part B?

Answer 25

what is part B?

Answer 26

I can try my best with it:)

Answer 27

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

Answer 28

Oh, good

Answer 29

```
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.

Answer 30

so (5,1000)?

Answer 31

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.

Answer 32

hope that is fair.

Answer 33

yes

Answer 34

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

Answer 35

yes, that is correct

Answer 36

And quite quick

Answer 37

1 graphed it

Answer 38

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,

Answer 39

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

Answer 40

okay

Answer 41

yes part C is correct

Answer 42

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

Answer 43

thats the maximum about of profit the promoter can make?

Answer 44

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

Answer 45

1000 makes more sense 1 think

Answer 46

yes

Answer 47

and what does 5 mean, can you tell me?

Answer 48

the amount the tickets cost?

Answer 49

yes

Answer 50

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

Answer 51

makes sense

Answer 52

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....

Answer 53

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

Answer 54

nope thats it

Answer 55

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

Answer 56

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

Answer 57

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