What is the optimal ticket price for a movie theater to maximize revenue?

[physicsgal]

1) "find the vertex algebraically"

y= -x^2 - 3
y = (x - 3)(x+1)??
(the -x is throwing me off)

2) a movie theatre sells tickets for $8.5 each. they are considering raising the prices but know that for every 50 cents the price is raised, 20 fewer people go to the movies. R = -40c^2 + 84c
c = ticket cost
R = revenue

what should the theatre charge to maximize revenue?

for this one i can probably just use my graphing calculator to find the roots?. but what values should i put in the window? (new to using graphing calculator)

~Amy

 

[courtrigrad]

1. The vertex is $$(-\frac{b}{2a}, f(-\frac{b}{2a}))$$. So $$y = -x^{2} - 3$$, $$b = 0$$. Therefore, the vertex is $$(0, f(0))$$ or $$(0,-3)$$.2. Find the vertex of $$R(c).$$ $$b = 84, a = -40$$.

 

[physicsgal]

thanks!

for 2) if i do the quadratic formula i end up with -85, and -82.95.. does this mean anything?

~Amy

 

[courtrigrad]

Since $$R(c)$$ is quadratic, then the maximum revenue would occur at the vertex, not at the roots. So it would be $$(\frac{-84}{-80}, 44.1)$$ or $$(1.05, 44.1)$$.

 

[physicsgal]

thanks! where did the 46.2 come from?

~Amy

 

[courtrigrad]

$$R(1.05) = -40(1.05)^{2} + 84(1.05) = 44.1$$. Should not be 46.2. My bad.

 

[physicsgal]

actually does that answer the question about whether or not they should raise the ticket price or not? I am missing more pieces of this puzzle.

they should increase the price to $44?

~Amy

 

[courtrigrad]

they should charge $1.05 according to the revenue function. Is there more information to this question?

 

[physicsgal]

that's all the info to the question. but the answer sounds a bit iffy. thanks, but i dunno. if you calculate $1.50 for c into the original formula:
R = -40c^2 + 84c
the revenue = 36

unless I am calculating that wrong

~Amy

 

[courtrigrad]

Its $1.05 not $1.50.

 

[physicsgal]

k, with $1.05 i calculated the revenue to be $1852.2

with the original $8.5 i calculated it to be $116314

~Amy

 

[courtrigrad]

Are you sure? In $$R(c) = -40c^2 + 84c$$, the -40 is outside of the squared expression. So you square c first, then multiply it by -40 and add it to 84c.

$$-40c^2$$ does not equal $$(-40c)^{2}$$

 

[physicsgal]

k, sorry for the hassle, looks like you are right

thanks again!

~Amy