Question

The student council is planning a dance for their high school. They did some research and found that the relationship between the ticket price and income that they will receive from the dance can be modeled by the function f(x) = ‒100(x ‒ 3)2 + 1500. They also calculated their expenses and found that their expenses can be modeled by the function g(x) = 300 + 10x. What ticket price(s) could the student council charge for the dance if they wanted to break-even (the expenses are equal to the income)?

Answer 1

To find the ticket price that will break even, we can set the total income equal to the total expenses by setting the two given functions equal to each other and solving for x.

Step-by-step explanation:

To find the ticket price that will break even, we need to set the total income equal to the total expenses. The total income is given by the function f(x) = ‒100(x ‒ 3)^2 + 1500, and the total expenses are given by the function g(x) = 300 + 10x. We can set these two functions equal to each other and solve for x.

f(x) = g(x)

‒100(x ‒ 3)^2 + 1500 = 300 + 10x

Simplifying the equation:

‒100(x^2 ‒ 6x + 9) + 1500 = 300 + 10x

‒100x^2 + 600x ‒ 900 + 1500 = 300 + 10x

‒100x^2 + 600x + 600 = 10x + 300

‒100x^2 + 590x + 300 = 0

Now we can solve this quadratic equation using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Using the coefficients of the quadratic equation, a = -100, b = 590, c = 300, we can substitute these values into the quadratic formula to find the possible ticket prices that will break even.