Find the break-even point for the firm whose cost function C and revenue function R are given.

C(x) = 180x + 12,000; R(x) = 270x

To find the break-even point, we need to determine the value of x where the cost function C(x) is equal to the revenue function R(x).

1. Set the cost function equal to the revenue function:
180x + 12,000 = 270x

2. Solve for x by isolating the variable:
12,000 = 90x
x = 12,000 / 90
x = 133.33

Therefore, the break-even point for the firm occurs when x is approximately equal to 133.33 units.