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

Set them equal or R(x) - C(x) = 0

2x + 10 = -5x^2 + 21x
5x^2 - 19x + 10 = 0
x = .6311 thousand or appr 631
or
x = 3.169 thousand or appr 3169

check:
if x = .6311 , R(.6311) - C(.6311) = .00053.. , not bad
if x = 3.169 , ...... = .001805

both answers are valid

To break even, the revenue (R(x)) should be equal to the cost (C(x)).

Given that R(x) = -5x^2 + 21x and C(x) = 2x + 10, we need to set these two functions equal to each other:

-5x^2 + 21x = 2x + 10

Now, we can solve for x by rearranging the equation and bringing all terms to one side:

-5x^2 + 21x - 2x - 10 = 0

Simplifying the equation gives:

-5x^2 + 19x - 10 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = -5, b = 19, and c = -10. Plugging in these values, we get:

x = (-(19) ± √(19^2 - 4(-5)(-10))) / (2(-5))

Simplifying further:

x = (-19 ± √(361 - 200)) / (-10)

x = (-19 ± √(161)) / (-10)

Now, let's evaluate both solutions:

x = (-19 + √161) / (-10) ≈ 0.99

x = (-19 - √161) / (-10) ≈ 3.04

Since the number of pairs of shorts cannot be negative, we can discard the solution x ≈ 3.04.

Therefore, the company must sell approximately 0.99 thousand (or 990) pairs of shorts to break even.

To determine the number of pairs of shorts the company must sell to break even, we need to find the point at which the revenue equals the cost.

The revenue function is given by R(x) = -5x^2 + 21x and the cost function is C(x) = 2x + 10.

To find the break-even point, we need to set the revenue equal to the cost and solve for x:

-5x^2 + 21x = 2x + 10

Let's rearrange the equation to set it equal to zero:

-5x^2 + 19x - 10 = 0

Now, we can solve this quadratic equation. We can factor it, complete the square, or use the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = -5, b = 19, and c = -10. Plugging in these values, we get:

x = (-19 ± √(19^2 - 4(-5)(-10))) / (2(-5))

Simplifying further:

x = (-19 ± √(361 - 200)) / -10
x = (-19 ± √161) / -10

Now we have two possible solutions:

x1 = (-19 + √161) / -10
x2 = (-19 - √161) / -10

Since it doesn't make sense to have a negative number of pairs of shorts, we can ignore the negative solution. Therefore, the number of pairs of shorts the company must sell to break even is:

x = (-19 + √161) / -10

You can plug this into a calculator for the exact value.