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?

let the number of 25 cent increases by x

income = ticket price x number of tickets sold

income = (6+.25x)(1200-25x)
= 7200 - 150x + 300x - 6.25x^2
= -6.25x^2 150x + 7200

d(income)/dx = -12.5x + 150 = 0 for a max of income
12.5x = 150
x = 12

so the price for max income should be 6 + .25(12) = $9.00

To find the price that would maximize the income, we need to determine the ticket price at which the total revenue is highest.

First, let's analyze the given information:
- The auditorium has 1600 seats.
- At a ticket price of $6, the owners can sell 1200 tickets.

Now, let's calculate the revenue at the current price of $6:
Total Revenue = Ticket Price * Number of Tickets Sold
Total Revenue at $6 = $6 * 1200 = $7200

Next, we know that for each 25 cent increase in price, the number of tickets sold reduces by 25. This means that if the ticket price is increased to $6.25, the owners would only be able to sell 1175 tickets.

Now, let's calculate the revenue at $6.25:
Total Revenue at $6.25 = $6.25 * 1175 = $7343.75

By comparing the revenues at $6 and $6.25, we can see that increasing the price by 25 cents increased the revenue.

Continuing this comparison, we can increase the price to $6.50 and calculate the revenue at that price:
Total Revenue at $6.50 = $6.50 * (1175 - 25) = $7481.25

We can observe that increasing the price to $6.50 further increases the revenue.

Continuing this process, we can calculate revenues at $6.75, $7.00, and so on, until we find a ticket price that results in a decrease in revenue.

At some point, the decreasing number of tickets sold due to the increasing ticket price will cause the revenue to start decreasing.

To summarize, we need to calculate the revenue at each ticket price increment of $0.25 until the point where the revenue starts decreasing. The ticket price where the revenue is highest will be the price that the owners should set to maximize their income.