Question

6.) 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+3,250n-9,000

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 )?

Answer 1

You can see that as n gets very large, profit becomes negative. Profit is also negative when n is zero. The nature of this second degree (n^2) polynomial is a parabola. It has only one maximum when the derivative w.r.t. n is equal to zero.

This occurs when 0 = -500n+3250

Answer 2

@Compassionate
@tHe_FiZiCx99
@Zale101

Answer 3

You are given the function P(n) = -250n^2+3,250n-9,000.

Answer 4

In Part A you are asked to find the zeroes of the function, right?

Answer 5

oh I got part a already

Answer 6

I factored 0 = -250n^2+3,250n-9,000
and got -250(n-4) (n-9)
which is the zero’s or solution of the function which represents the x intercepts.

Answer 7

Okay, Hang on, let me check that

Answer 8

is this the one you asked for my help on?

Answer 9

Okay, that looks good. In Part B, you are asked to find the maximum profit by completing the square, right?

Answer 10

yes

Answer 11

Okay, so to complete the square, we have to first put P(n) in the form Ax^2 + Bx = C right?

Answer 12

yeah

Answer 13

The first step is to Set P(n) = 0 so

0 = -250n^2+3,250n-9,000

Answer 14

Then add 9000 to both sides to get

9000 = -250n^2 + 3250n

which is the same as

-250n^2 + 3250n = 9000

Notice that has the same form as
Ax^2 + Bx = C

Answer 15

Actually, if we divide both sides by -1 we get
250n^2 - 3250n = -9000

Now we have it in the form
Ax^2 + Bx = C

Answer 16

Now we need to divide both sides by 250. After doing so we get

n^2 - 13n = -36

Answer 17

At this point we have it in a form where we can complete the square. Do you agree?

Answer 18

yes

Answer 19

Do you remember what needs to be done in order to complete the square? What should we add to both sides?

Answer 20

you add a needed value to each side to balance the equation out @Hero

Answer 21

Once you have it in the form x^2 + bx = c, you complete the square by adding \(\left(\frac{b}{2}\right)^2\) to both sides.

Answer 22

In this case, b = -13 so we will add \(\left(\frac{-13}{2}\right)^2\) to both sides of \(n^2 - 13n = -36\).

Answer 23

Upon doing so we will have:

\(n^2 - 13n + \left(\frac{-13}{2}\right)^2 = \left(\frac{-13}{2}\right)^2 - 36\)

Answer 24

Do you agree?

Answer 25

yeah

Answer 26

Do you know what that simplifes to?

Answer 27

simplify (-13/2)^2-36 to 25/4?

Answer 28

right @Hero

Answer 29

Yes, you correctly simplified the right side. What about the left side?

Answer 30

We need to write the left side as a binomial square.

Answer 31

n^2-13n- (13/2)?

Answer 32

Do you remember how to expand\(\left(\frac{-13}{2}\right)^2\)?

Answer 33

nope

Answer 34

Hint:

\(\left(\dfrac{a}{b}\right)^2 = \dfrac{a^2}{b^2}\)

Answer 35

When you expand it, you get 169/4. That was correct.

Answer 36

oh okay wasn't for sure

Answer 37

However, I remembered, that you don't need to expand it on the left side because basically

\(n^2 - 13n + \left(\frac{-13}{2}\right)^2\) = \(\left(n - \frac{13}{2}\right)^2\)

Answer 38

So what you end up with is

\(\left(n - \frac{13}{2}\right)^2 = \frac{25}{4}\)

Answer 39

We have to keep solving for n, so what do you think we should to next?

Answer 40

well I know we have to iscolate the variable so im thinking we subtract from the side with the variable

Answer 41

@Hero

Answer 42

Actually, first we have to take the square root of both sides:

\(\sqrt{\left(n - \frac{13}{2}\right)^2} = \sqrt{\frac{25}{4}}\)

Answer 43

What do you know about the relationship between a square and square root?

Answer 44

not much, ive just heard of them that's it

Answer 45

why @Hero

Answer 46

n= 4 and 9 @Hero

Answer 47

is that right

Answer 48

You solved for n the rest of the way? Can you show the steps you took to find n?