If a company sells its product to distributors in boxes of 10 units each. The company needs to sell 200 boxes in order to break even.
What is the break even?
Set the profit equation equal to 0 and solve for n.
The profit equation is given as p = -n² + 300n + 100,000
Setting p = 0 we have 0 = -n² + 300n + 100,000
To solve this quadratic equation use the quadratic formula:
n = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation.
a = -1, b = 300, and c = 100,000
Plugging in the formula:
n = (-(300) ± √((300)² - 4(-1)(100,000))) / (2(-1))
n = (-300 ± √(90,000 + 400,000)) / (-2)
n = (-300 ± √(490,000)) / (-2)
n = (-300 ± 700) / (-2)
Simplify the expression:
n = (400) / 2 or n = (-1000) / 2
n = 200 or n = -500
We ignore the negative solution since, in this case the number of boxes cannot be negative.
Therefore the company needs to sell 200 boxes in order to break even.