Quadratic Function Algebra Problem

Question

A premium movie theater currently charges $14.00 per ticket and sells 900 tickets every week. 

A) What is the current weekly revenue? 

B) The theater estimates that for each $0.50 increase in ticket price, the number of tickets sold decreases by 60. Write down a linear function that determines the number of tickets sold, qq, as a linear function of the ticket price, x. [Hint: determine some (x, q) points and then use those to find the slope and come up with the linear function] 

C) Set up the quadratic function that models the revenue of the theater as a function of the ticket price, x. (You should be incorporating your answer from part B). 

D) What price will maximize ticket revenue for the theater? How many tickets are sold at that price? What is the maximum revenue? You may show your work either graphically or Algebraically.

Answer 1

Hello, Monica,

Let q be tickets sold in a week and x is price per ticket. The standard form of a linear equation is y=mx+b, where m is the slope and b is the y-intercept (the value of y when x is zero). Since the question wants us to use q for weekly tickets sold, we can rewrite this as q = mx + b.

A) The weekly revenue would be the number of tickets sold in a week times the price per ticket: (900/week)*($14/ticket) = $12,600/week

B) One can find a linear equation either by graphing a few points and finding the slope and y-intercept from the graph, or deduce both from a few data points taken from the same table. Make a table with a few points calculated from the knowledge that 60 fewer tickets are sold each week (q) when the price (x) increases by $0.50/ticket:

We can take any two points and calculate the slope, m. I'll pick (14,900) and (15,780). The slope is the "rise over the run," or change in the y axis (q) per a change on the x axis. Rise = (780-900) = -120. Run is (15-14)=1, therefore Rise/Run = -120. The slope, m, is -120. [Doesn't sound good, but that's business when you raise prices]. The linear equation, q = mx + b, is now q = -120x + b. We need to find b, the y-intercept. Do that by entwering any of the data points and solve for b. I'll use (14,900):

q = -120x + b

900 = -120*(14) + b

900=-1680 + b

b = 2580 [This is saying that 2580 tickets will be "sold" when the price is zero. Silly, but that is math]

So the full equation becomes q=-120x + 2580

You can also do this by graphing a few of the points in the list and finding the data for slope and y-intercept from the graph.

C)

We now want the total revenue as a function of ticket price. Total revenue, R, would be the tickets sold per week times the price per ticket. Use the linear equation above for q.

Total revenue(R) = q*x

R = (-120x + 2580)*(x)

R = -120x² + 2580x

We can plot this parabola and find the vertex, os we can take the first ferivative and set it equal to zero, the slope at the vertex. I'll try the first derivation:

R = -120x² + 2580x

R' = -240x + 2580

0 = -240x + 2580

x = 10.75

The slope of the equation at 10.75 is zero, so this is the point that the "premium" movie theater will maximize profit. At $10.75/ticket, total revenue would be $13,867.5/week. [Not counting popcorn]

The question wants the number of tickets sold at this price, so use the linear equation we first developed:

q=-120x + 2580

q = -120(10.75)+2580

q = 1290 tickets/week

Bob