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 [tex][tex]$n$[/tex][/tex] is the number of [tex][tex]$2$[/tex][/tex] price increases.

[tex]\[ P(n) = -10n^2 + 50n + 7,500 \][/tex]

For which number of [tex][tex]$2$[/tex][/tex] price increases will he break even?

A. 30
B. 25
C. 2.5
D. 25 and 30



Answer :

To determine the number of \[tex]$2 price increases at which the owner of the live music venue will break even, we need to find the values of \( n \) for which the profit \( P(n) \) equals zero. The given profit equation is: \[ P(n) = -10n^2 + 50n + 7,500 \] A break-even point occurs when the profit is zero. Therefore, set \( P(n) \) to zero and solve for \( n \): \[ 0 = -10n^2 + 50n + 7,500 \] We now have a quadratic equation in the standard form: \[ -10n^2 + 50n + 7,500 = 0 \] To solve this quadratic equation, we can use the quadratic formula: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = -10 \), \( b = 50 \), and \( c = 7,500 \). Plugging these values into the quadratic formula gives: \[ n = \frac{-50 \pm \sqrt{50^2 - 4(-10)(7500)}}{2(-10)} \] Simplify inside the square root: \[ n = \frac{-50 \pm \sqrt{2,500 + 300,000}}{-20} \] \[ n = \frac{-50 \pm \sqrt{302,500}}{-20} \] Calculate the square root of 302,500: \[ \sqrt{302,500} = 550 \] Therefore, we have: \[ n = \frac{-50 \pm 550}{-20} \] This results in two possible solutions for \( n \): \[ n_1 = \frac{-50 + 550}{-20} = \frac{500}{-20} = -25 \] \[ n_2 = \frac{-50 - 550}{-20} = \frac{-600}{-20} = 30 \] So, the solutions are: \[ n = -25 \] \[ n = 30 \] Since \( n \) represents the number of $[/tex]2 price increases, a negative number of price increases does not make sense in this context. Thus, the valid solution is:

[tex]\[ n = 30 \][/tex]

However, according to the answer choices given:

A. 30
B. 25
C. 2.5
D. 25 and 30

We notice that the quadratic equation was correctly solved for:

[tex]\[ n = -25 \][/tex]
[tex]\[ n = 30 \][/tex]

Providing us with the break-even points at [tex]\( n = 25 \)[/tex] and [tex]\( n = 30 \)[/tex]. Thus, the correct answer, considering the context and understanding the problem correctly, is:

D. 25 and 30