#### Answer

(a) $C(x) = 100x + 100,000$
(b) $R(x) = 300x$
(c) $x = 500$; the company must manufacture and sell 500 bicycles in order to reach the break-even point, which is to say, that the costs are fully offset by the revenue earned from selling bicycles.

#### Work Step by Step

(a) The exercise describes that the company has a fixed cost of 100,000 dollars PLUS 100 dollars per bicycle. This means that the total cost of production for the company is: $$C(x) = 100,000 + 100x$$ where $C(x)$ is the Cost function of bicycle manufacture and $x$ is the amount of units sold.
(b) The exercise also describes that the company sells each bicycle at a price of 300 dollars, which means that the total revenue produced by the company is: $$R(x) = 300x$$ where $R(x)$ is the Revenue function of selling bicycles
(c) The break-even point is the point where the costs of bicycle manufacture are offset by the revenue earned from selling them. This point is reached, then, when: $$C(x) = R(x)$$ $$100,000 + 100x = 300x$$ $$100, 000 = 200x$$ $$\frac{100,000}{200} = 500 = x$$ Therefore, the break-even point is reached when the company manufactures and sells 500 bicycles.