At a price of $7.5 per ticket, a musical theater group can fill every seat in their 1550 seat performance hall. For every additional dollar charged for admission, the number of tickets sold drops by 55.
a) What ticket price maximizes revenue? Round your answer to the nearest cent.
price = $
b) How many seats are sold at that price? Round your answer to the nearest whole number.
number of seats sold =
To find the ticket price that maximizes revenue, we need to determine the revenue function based on the given information.
Let's begin by figuring out the number of tickets sold at a specific price. We know that for every additional dollar charged for admission, the number of tickets sold drops by 55.
Let's assume the ticket price is x dollars. Since the initial price is $7.5 per ticket, the number of dollars added to the initial price is (x - 7.5).
So the number of tickets sold at a price of x dollars is given by:
Number of Tickets Sold = 1550 - 55(x - 7.5)
Now, the revenue function is calculated by multiplying the ticket price by the number of tickets sold:
Revenue = x * (1550 - 55(x - 7.5))
a) To determine the ticket price that maximizes revenue, we need to find the value of x when the revenue function is maximized. This can be done by finding the vertex of the revenue function, which represents the highest point on the graph.
To find the vertex, we can use the formula x = -b / (2a) for a quadratic equation in the form ax^2 + bx + c = 0.
In our case, the quadratic equation is in the form -55x^2 + 4200x - 11625 = 0. Comparing this to the quadratic equation ax^2 + bx + c = 0, we can see that a = -55, b = 4200, and c = -11625.
Using the formula, the x-coordinate of the vertex is given by:
x = -4200 / (2 * -55) = 381.82
Rounding this to the nearest cent, the ticket price that maximizes revenue is approximately $381.82.
b) To find the number of seats sold at that price, we substitute the value of x (381.82) into the equation for the number of tickets sold:
Number of Seats Sold = 1550 - 55(381.82 - 7.5)
Evaluating this expression, we find that the number of seats sold at the price of $381.82 is approximately 171.