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.
