At a price of $7.5 per ticket, a musical theater group can fill every seat in their 1500 seat performance hall. For every additional dollar charged for admission, the number of tickets sold drops by 95.
a) What ticket price maximizes revenue? Round your answer to the nearest cent.
price = $
b) How many seats are sold at that price? Round your answer to the nearest whole number.
number of seats sold =
To find the ticket price that maximizes revenue, we need to determine the price at which the product of the price and the number of seats sold is highest.
Let's break it down into steps:
1. Start with the base price of $7.5 per ticket.
2. For each additional dollar charged for admission, the number of tickets sold drops by 95.
3. So, if the ticket price is increased by $1, the number of tickets sold will be 1-95 = 1505 (1500 - 95).
4. We can generalize this relationship as follows:
- If the ticket price is $7.5, the number of tickets sold is 1500.
- If the ticket price is $8.5 ($7.5 + $1), the number of tickets sold is 1500 - 95 = 1505.
- If the ticket price is $9.5 ($7.5 + $2), the number of tickets sold is 1500 - 2 * 95 = 1410, and so on.
Now, let's calculate the revenue for each price option and find the maximum:
Price = $7.5, Number of seats sold = 1500, Revenue = $7.5 * 1500 = $11250
Price = $8.5, Number of seats sold = 1505, Revenue = $8.5 * 1505 = $12792.5
Price = $9.5, Number of seats sold = 1410, Revenue = $9.5 * 1410 = $13395
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Continue this calculation for multiple price options and determine the revenue for each. The price that gives the highest revenue is the one that maximizes revenue. Round the answer to the nearest cent.
Calculate until you find the maximum revenue. Take note of the ticket price and the number of seats sold.
Once you have found the ticket price that maximizes revenue in part a), you can determine the number of seats sold at that price. Round the answer to the nearest whole number.