A movie theater wants to determine the price per ticket that should be charged in order to maximize its
revenue. The theater can accommodate 2000 movie goers per night, but the average attendance is about 1400
people per night. The average ticket price is about $8.00. Market research shows that, for each $0.50
reduction in the ticket price, attendance increases by 250 spectators. What should the theater charge per ticket
in order to sell out every show?
I figured out the answer!
To determine the price per ticket that the theater should charge in order to sell out every show and maximize its revenue, we can follow these steps:
Step 1: Calculate the current revenue per night.
The average attendance is 1400 people per night, and the average ticket price is $8.00.
Revenue per night = Average attendance * Average ticket price
Revenue per night = 1400 * $8.00
Step 2: Calculate the potential increase in attendance for every $0.50 reduction in ticket price.
The market research shows that for each $0.50 reduction in the ticket price, attendance increases by 250 spectators.
Step 3: Calculate the number of $0.50 reductions needed to reach maximum attendance.
To sell out every show, the theater needs to reach its maximum attendance of 2000 people per night. So, we need to calculate how many reductions in ticket price are needed to attract the additional 600 spectators.
Number of reductions = (Maximum attendance - Average attendance) / Increased attendance per reduction
Number of reductions = (2000 - 1400) / 250
Step 4: Calculate the new ticket price.
To find the new ticket price, we multiply the number of reductions by $0.50 and subtract it from the current ticket price.
New ticket price = Current ticket price - (Number of reductions * $0.50)
New ticket price = $8.00 - (Number of reductions * $0.50)
Step 5: Calculate the maximum revenue per night with the adjusted ticket price.
Maximum revenue per night = Maximum attendance * New ticket price
Maximum revenue per night = 2000 * New ticket price
By following these steps, the theater can determine the price per ticket it should charge in order to sell out every show and maximize its revenue.
To determine the price per ticket that should be charged in order to maximize revenue, we need to consider the relationship between price and attendance.
Based on the given information, we know that reducing the ticket price by $0.50 leads to an increase in attendance by 250 spectators. This implies that there is a linear relationship between attendance and ticket price.
Let's break down the problem into steps:
1. Calculate the maximum attendance possible: Since the theater can accommodate 2000 moviegoers per night, the maximum attendance is 2000.
2. Determine the change in attendance for each $0.50 reduction in ticket price: Since the attendance increases by 250 spectators for each $0.50 reduction, we can calculate the change per unit price. The change in attendance per unit price is given by 250/0.50 = 500. So, for every $1 decrease in ticket price, attendance will increase by 500.
3. Find the revenue for different ticket prices: Revenue is calculated by multiplying the ticket price by the attendance. Let's assume the ticket price to be x dollars. The attendance can be calculated as (1400 + 500(x-8)), where x-8 represents the number of $1 decreases in ticket price from the original price of $8. The revenue can be calculated as x * (1400 + 500(x-8)).
4. Find the ticket price that maximizes the revenue: We can express the revenue function as R(x) = x(1400 + 500(x-8)). To find the ticket price that maximizes the revenue, we need to find the value of x that gives the maximum value of R(x). This can be achieved by finding the derivative of R(x) with respect to x and setting it equal to zero. Differentiating and solving the equation will provide the value of x that optimizes the revenue.
By following these steps, we can determine the ticket price per ticket that should be charged in order to maximize the theater's revenue.