Kool Klothes has determined that the revenue function for selling x thousand pairs of shorts is R(x) =-5x^2 + 21x. The cost function C(x) = 2x + 10 is the cost of producing the shorts.
How many pairs of shorts must the company sell in order to break even
you want cost = revenue
2x+10 = -5x^2+21x
5x^2 - 19x + 10 = 0
I think that R(x) is bogus, since it starts to decline as more shorts are sold.
http://www.wolframalpha.com/input/?i=2x%2B10+%3C+-5x%5E2%2B21x
To find the break-even point, we need to find the value of x where the revenue (R) equals the cost (C).
The revenue function is R(x) = -5x^2 + 21x, and the cost function is C(x) = 2x + 10.
To find the break-even point, we need to set R(x) equal to C(x):
-5x^2 + 21x = 2x + 10
We can solve this quadratic equation by moving all the terms to one side:
-5x^2 + 21x - 2x - 10 = 0
-5x^2 + 19x - 10 = 0
Now, we can factor this quadratic equation or use the quadratic formula to solve for x. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -5, b = 19, and c = -10:
x = (-19 ± √(19^2 - 4*(-5)*(-10))) / (2*(-5))
Simplifying this:
x = (-19 ± √(361 - 200)) / (-10)
x = (-19 ± √161) / (-10)
Now, we have two possible solutions, one with the plus sign and one with the minus sign:
x1 = (-19 + √161) / (-10)
x2 = (-19 - √161) / (-10)
Calculating these values:
x1 ≈ 0.33 (rounded to two decimal places)
x2 ≈ 3.07 (rounded to two decimal places)
Since the number of shorts must be a whole number, Kool Klothes must sell either 0 or 4 pairs of shorts to break even.