anonymous one year ago 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)

1. anonymous

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

2. anonymous

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

3. anonymous

no that is wrong

4. anonymous

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

5. anonymous

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.

6. anonymous

is that correct?

7. anonymous

yes looks good to me

8. anonymous

Okay (:

9. anonymous

Now how do I "complete the square"

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

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

14. anonymous

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

15. anonymous

factor out the $$-250$$ from the first two terms

16. anonymous

$-250n^2 + 2,500n - 4,000$ $-250(n^2-10n)-4,000$

17. anonymous

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

18. anonymous

Okay i got that aha

19. anonymous

right that is step one

20. anonymous

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

21. anonymous

Yeah... I got lost there ):

22. anonymous

yeah i can see why

23. anonymous

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

24. anonymous

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

25. anonymous

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

26. anonymous

why 25 though?

27. anonymous

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

28. anonymous

Ohhh

29. anonymous

Okay (:

30. anonymous

now don't forget the $$-250$$ outside the parentheses

31. anonymous

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

32. anonymous

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

33. anonymous

but why the times symbol?

34. anonymous

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

35. anonymous

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

36. anonymous

I meant the times symbol for 25 x 250

37. anonymous

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

38. anonymous

you subtracted $$6250$$ from the original expression, so you have to add it back

39. anonymous

there is really a much much easier way to do it

40. anonymous

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

41. anonymous

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 $$P(5)$$

42. anonymous

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

43. anonymous

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

44. anonymous

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

45. anonymous

compute the number at the end

46. anonymous

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

47. anonymous

$-400+25\times 250=2250$

48. anonymous

-250(n-5)^2+2250

49. anonymous

yes

50. anonymous

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

51. anonymous

@perl