Maximize Revenue for Quadratic Problem at Movie Theatre

[Imperil]

A movie theatre sells tickets for $8.50 each. The manager is considering raising the prices but knows that for every 50 cents the price is raised, 20 fewer people go to the movies. The equation R = -40c^2 = 720c describes the relationship between the cost of tickets, c dollars, and the amount of revenue, R dollars, that the theatre makes. What price should the theatre charge to maximize revenue?

I believe what I need to do is find the maximum vertex of the parabola in order to solve the equation. So I did the following:

R = -40c^2 - 720c
= -40(c^2 - 18c)
= -40(c^2 - 18c + 9^2 - 9^2) <-- complete the square
= -40(c^2 - 18c + 81 - 81)
= -40[(c^2 - 9)^2 - 81)
= -40(c^2 - 9)^2 + 3240

Which would give me a vertex (9, 3240) but this does not make sense to me, I am not sure what I am looking for to be honest. I believe that the maximum price would be $9.00 to have a revenue of $3240, is this correct and I am just second guessing?

 

[symbolipoint]

You seem to be thinking in the right direction, although I did not analyze your work in detail. One spot of confusion is what you say, equation R = -40c^2 = 720c describes the relationship between the cost of tickets, c dollars, and the amount of revenue, R dollars, that the theatre makes", does not make sense. OOOOHHH, you mean -40c^2 - 720c = R, this could be better.

 

[nickjer]

R = -40c^2 - 720c
= -40(c^2 - 18c)

You pulled out a negative but you left the 2nd term negative as well.

Double check the equation you were given, because you miswrote it in the problem, and it could have a mistake when you first started solving it.

 

[Imperil]

Now I am fairly confused as it really does not make sense to me. I double checked the equation and I was correct in my work that it is the following:

R = -40c^2 - 720c

After correcting my mistake (that was pointed out by nickjer) I now have the following:

R = -40c^2 - 720c
= -40(c^2 + 18c)
= -40(c^2 + 18c + 81 - 81) <-- complete the square
= -40[(c^2 + 9)^2 - 81]
= -40(c^2 + 9)^2 + 3240

Which would give a vertex of (-9, 3240) which makes no sense to me in the context of the question. I am really not sure where to go from here.

 

[gabbagabbahey]

Now I am fairly confused as it really does not make sense to me. I double checked the equation and I was correct in my work that it is the following:

R = -40c^2 - 720c

Surely, this equation should be R=-40c^2+720c instead!

 

[Imperil]

I have triple checked and it is definitely -720c which is why I am confused.

 

[gabbagabbahey]

I have triple checked and it is definitely -720c which is why I am confused.

It must be a typo!

If the equation were -40c^2-720c , then if you charged $1.00 per ticket, you would have a revenue of -$760.00; but revenue is always a positive quantity.

I would assume that the equation is supposed to be -40c^2+720c and just ask your instructor about it when you see him/her.

 

[Imperil]

I thought this exact same thing but figured maybe I was thinking about it wrong! Thanks for your help, I just contacted my teacher by email regarding this. It is a key problem in my correspondence that I need to hand in, so I am shocked they included this typo.