Question

Fan and Medal!

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 - 4,000

Part A: What are the zeroes of the above function, and what do they represent? Show your work. (4 points)

Part B: Find the maximum profit by completing the square of the function P(n). Show the steps of your work. (4 points)

Part C: What is the axis of symmetry of the function P(n)? (2 points)

Answer 1

\[P(n) = -250n^2 + 2,500n - 4,000 \] you want the zeros?

Answer 2

I already finished part A
I know the 0's are 2 and 8

Answer 3

no that is wrong

Answer 4

\(n\) is the price
i guess if he prices the tickets at $2 or at $8 he makes no money

Answer 5

The zeroes of the function would be 2 and 8. The zeroes of the function above represents the specific prices, 2 and 8, which will result in a zero profit.

Answer 6

is that correct?

Answer 7

yes looks good to me

Answer 8

Okay (:

Answer 9

Now how do I "complete the square"

Answer 10

there is a much easier way to do it (find the max) than completing the square, but we can do it that way too if you have to

Answer 11

first of all it is pretty obvious that if it is zero at 2 and at 8 then it is the biggest half way between at 5

Answer 12

plus the first coordinate of the vertex is always
\[-\frac{b}{2a}\] which in your case is
\[-\frac{2500}{2\times (-250)}=5\]

Answer 13

Yeah, but it specifically asks to complete the square (which I hate because it's so confusing)

Answer 14

but we can still complete the square if you are dying to do it

Answer 15

factor out the \(-250\) from the first two terms

Answer 16

\[ -250n^2 + 2,500n - 4,000 \]
\[-250(n^2-10n)-4,000\]

Answer 17

P(n)=-250(n^2-10n)-4000?

Answer 18

Okay i got that aha

Answer 19

right that is step one

Answer 20

then half of \(-10\) is \(-5\) complete the square via
\[-250(n-5)^2-4000+25\times 250\]

Answer 21

Yeah... I got lost there ):

Answer 22

yeah i can see why

Answer 23

when you replace \(n^2-10n\) by \(n-5)^2\) you have changed it

Answer 24

\[(n-5)^2=n^2-10n+25\] so you have added \(25\) when you make the change

Answer 25

Ohh, okay I see that now, but like if it's so many numbers and such, my brain likes to just stop reading it

Answer 26

why 25 though?

Answer 27

because when you square \(n-5\) you get \((n-5)^2=n^2-10n+25\)

Answer 28

Ohhh

Answer 29

Okay (:

Answer 30

now don't forget the \(-250\) outside the parentheses

Answer 31

so actually you have subtracted \(25\times 250\) so you have to add it back

Answer 32

hence \[-250(n-5)^2-4000+25\times 250\]

Answer 33

but why the times symbol?

Answer 34

what else would you like me to write? look at this line \[-250(n-5)^2\]

Answer 35

if you multiply that out you get
\[-250(n^2-10n+25)\] as a first step

Answer 36

I meant the times symbol for 25 x 250

Answer 37

and when you multiply that out, you get \[-250n^2+2500n-6250\]

Answer 38

you subtracted \(6250\) from the original expression, so you have to add it back

Answer 39

there is really a much much easier way to do it

Answer 40

Yeah, but my algebra teachers just love making us suffer haha

Answer 41

since you know that the first coordinate of the vertex is \(5\) if you want the second coordinate, all you have to do is find \(P(5)\)

Answer 42

you can pretend to to it the other way, but then just find \(P(5)\) and write that out at the end

Answer 43

so after P(n)=-250(n-5)^2-4000+25*250 ,what do I do, or is that the final step?

Answer 44

algebra teachers lead sad lives, most are either frustrated mathematicians or are teaching at the outer edge of their knowledge

Answer 45

compute the number at the end

Answer 46

-250(n-5)^2-4000+6250

Answer 47

\[-400+25\times 250=2250\]

Answer 48

-250(n-5)^2+2250

Answer 49

yes

Answer 50

And now? do I make it (n-5)(n-5)?

Answer 51

@perl