Revenue is the product of ticket price and attendance. Since you are assuming that attendance is linearly related to attendance, you can also see that with every dollar drop in price, attendance increases by 3000. Conversely, with every dollar increase in price, attendance drops by 3000. You can now set up a revenue function as follows...
$rev = (12 + x)(22000 - 3000x)
In this function, (12 + x) = the ticket price with respect to x, and (22000 - 3000x) = the attendance with respect to x. As x increments by 1, the ticket price also increments by 1 while the attendance decrements by 3000, or conversely, if x decreases by 1, price will decrease by 1 while attendance increases by 3000. When you multiply the terms together, you get the quadratic function...
264000 - 14000x - 3000x2
When you find the y value of the vertex of this quadratic function, that will give you the max revenue. The x value of the vertex can be found using the formula -b/2a. In this case b = -14000 and a = -3000, so...
-b/2a = -(-14000)/2(-3000) = -7/3
Since you are asked to find the ticket price, you need to use the expression that was derived earlier for ticket price, which was (12 + x). When you plug -7/3 for x, you get a ticket price of $9.67
