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.
a) How many pairs of shorts must the company sell in order to break even
To break even, the revenue generated must equal the cost incurred. In this case, we need to find the value of x that satisfies the equation R(x) = C(x).
The revenue function is given as R(x) = -5x^2 + 21x, and the cost function is given as C(x) = 2x + 10.
To find the breakeven point, we need to set R(x) equal to C(x) and solve for x:
-5x^2 + 21x = 2x + 10
Rearranging the equation, we get:
-5x^2 + 19x - 10 = 0
This is a quadratic equation. To solve it, we can either factor, complete the square, or use the quadratic formula.
Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = -5, b = 19, and c = -10.
Plugging these values into the formula, we get:
x = (-19 ± sqrt(19^2 - 4(-5)(-10))) / (2(-5))
Simplifying further:
x = (-19 ± sqrt(361 - 200)) / (-10)
x = (-19 ± sqrt(161)) / (-10)
The square root of 161 is approximately 12.69, so we have:
x = (-19 + 12.69) / (-10) or x = (-19 - 12.69) / (-10)
Simplifying further:
x = -6.31 / (-10) or x = -31.69 / (-10)
x = 0.631 or x = 3.169
Since we are selling shorts in thousands, the company must sell approximately 0.631 thousand (or 631 pairs) or 3.169 thousand (or 3169 pairs) of shorts in order to break even.