Final answer:
To determine the ticket price that would maximize revenue, we can use the concept of linear demand. The revenue is given by the product of attendance and ticket price. By finding the vertex of the quadratic function representing revenue, we can determine the ticket price that maximizes revenue.
Step-by-step explanation:
To determine the ticket price that would maximize revenue, we can use the concept of linear demand. The demand for tickets can be represented by a linear equation of the form y = mx + b, where y is the attendance, x is the ticket price, m is the slope of the line, and b is the y-intercept.
We are given two points (27000, 11) and (32000, 8) that lie on the demand line. We can use these points to find the slope (m) of the line:
m = (32000 - 27000) / (8 - 11) = 5000 / (-3) = -1667
The equation of the demand line is then y = -1667x + b.
Revenue (R) is given by the product of attendance (y) and ticket price (x): R = xy.
We want to find the ticket price (x) that maximizes revenue (R). To do this, we can express revenue in terms of attendance: R = (-1667x + b)x.
Expanding this expression, we get R = -1667x^2 + bx. To find the ticket price that maximizes revenue, we can find the vertex of this quadratic function. Since the coefficient of x^2 is negative, the vertex will be the maximum point. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -1667 and b is the y-intercept.
The y-intercept can be found by substituting one of the points into the equation y = -1667x + b. Using the point (32000, 8), we get 8 = -1667(32000) + b, which gives b = 53336.
Substituting the values into the formula, we get x = -53336 / (2 * -1667) = 16.
Therefore, a ticket price of $16 would maximize revenue.