Question

The owner of a live music venue is considering changing his entrance fee. He can model the projected profits based on price increases using this

equation, where n is the number of $2 price increases.
P(n) = -10n2 + 50n + 7,500
For which nun er of $2 price increases will he break even?
A. 30
B. 2.5
C. 25
D. 25 and 30

Answer 1

The owner of a live music venue will break even at either 25 or 30 $2 price increases, according to the quadratic profit model provided.

Step-by-step explanation:

To determine the number of $2 price increases at which the owner of a live music venue will break even, we need to find the value of n that makes the profit equation P(n) = -10n2 + 50n + 7,500 equal to zero. To break even, the profits must be zero, meaning the revenue from ticket sales equals the costs of the event.

Setting the equation to zero gives us:

0 = -10n² + 50n + 7,500

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. However, a simple observation tells us that if we factor out 10, the equation simplifies to:

0 = -n² + 5n + 750

The factored form of the equation is (n - 25)(n - 30) = 0, giving us two potential break-even numbers of price increases, 25 and 30, which corresponds to option D.