Question 999839
first you want to create the formula for attendance.


this will be a straight line equation.


when x = 11, attendance is 13000
when x = 9, attendance is 15500


these are two coordinate points on your graph.


(x1,y1) is the first point.
(x2,y2) is the second point.


you have:


(x1,y1) = (11,13000)
(x2,y2) = (9,15500)


the slope intercept form of the equation for a straight line is y = mx + b.


m is the slope
b is the y-intercept.


given two points, m = (y2-y1) / (x2-x1).


this becomes (15500 - 13000) / (9 - 11) = 2500 / -2 = -1250


your slope is -1250.


y = mx + b becomes y = -1250x + b


to find b, take any coordinate point on the line and replace y with the y-coordinate and replace x with the x-coordinate and solve for b.


we'll use (x1,y1) = (11,13000)


y = -1250x + b becomes 13000 = -1250*11 + b


solve for b to get b = 13000 + 1250*11 = 26750


y = -1250x + b becomes y = -1250x + 26750.


that is the equation for attendance.


the graph of that equation is shown below:


<img src = "http://theo.x10hosting.com/2015/102903.jpg" alt="$$$" </>


based on the formula, you can see on the graph that:


when x = 0, attendance is 26750.
when x = 21.4, attendance is 0.
when x = 11, attendance is 13000.


not shown on the graph, but calculated from the formula, you also get:


when x = 9, attendance is 15000, confirming that the equation is good.


revenue is equal to attendance times price.


let y = revenue
let x = price
let -1250x + 26750 = attendance


equation becomes:


y = (-1250x + 26750) * x


you can graph this equation as is, or you can simplify it to get:


y = -1250x^2 + 26750x


you will get the same graph either way.


you can find the maximum value of y from the graph, or you can derive it from the quadratic equation formula of x = -b/2a and y = f(-b/2a)


in the equation of y = -1250x^2 + 26750x:


a = -1250
b = 26750


x = -b/2a = -26750 / -2500 = 10.7


f(-b/2a) becomes f(10.7)


replace x in the equation with 10.7 and you get:


y = -1250*10.7^2 + 26750*10.7 = 143112.5 which is shown as 143113 on the graph.


that is the maximum revenue that can be attained based on the formula.


the graph of the revenue equation is shown below:


<img src = "http://theo.x10hosting.com/2015/102904.jpg" alt="$$$" </>