An auditorium has 1600 seats. At a ticket price of $6, the owners can sell 1200 tickets. For each 25 cent increase in price the tickets sold reduces by 25. What price should the owners set to maximize the income?
To find the price that will maximize the income, we need to determine the ticket price that will generate the maximum revenue.
First, let's establish the relationship between ticket price, ticket sales, and revenue:
Revenue = Ticket Price x Ticket Sales
According to the given information, the auditorium has 1600 seats, and at a ticket price of $6, the owners can sell 1200 tickets. For every 25 cent increase in price, the ticket sales reduce by 25.
Let's break down the problem step by step:
1. Determine the initial ticket sales:
At a ticket price of $6, the owners can sell 1200 tickets.
2. Determine the initial revenue:
Revenue = Ticket Price x Ticket Sales
Revenue = $6 x 1200 = $7200
3. Determine the decrease in ticket sales:
For every 25 cent increase in price, the ticket sales reduce by 25.
In other words, for every $0.25 increase in ticket price, the ticket sales decrease by 25 tickets.
4. Determine the revenue at each price point:
Let's calculate the revenue at different price points to find the maximum:
- At $6.00 price:
Revenue = $6 x 1200 = $7200
- At $6.25 price:
Ticket sales = 1200 - 25 = 1175
Revenue = $6.25 x 1175 = $7343.75
- At $6.50 price:
Ticket sales = 1175 - 25 = 1150
Revenue = $6.50 x 1150 = $7475.00
- At $6.75 price:
Ticket sales = 1150 - 25 = 1125
Revenue = $6.75 x 1125 = $7593.75
By calculating the revenue at each price point, we can see that the revenue is increasing initially but starts to decrease after reaching a peak.
5. Find the maximum revenue:
From the calculations, we can observe that the maximum revenue occurs at a ticket price of $6.50, which generates a revenue of $7475.00.
Therefore, to maximize the income, the owners should set the ticket price at $6.50.