A company produces and sells hand-crafted desks. The price $p(x)$, in dollars per desk, when $x$ desks are produced and…

To break even, the company's profit must be $0, which means total revenue must equal total cost. Total revenue, $R(x)$, is the product of the number of desks sold, $x$, and the price per desk, $p(x)$: $R(x) = x(900 - 15x) = 900x - 15x²$. The profit function $P(x)$ is $R(x) - C(x) = (900x - 15x²) - (300x + 4{,}500) = -15x² + 600x - 4{,}500$. Setting $P(x) = 0$ gives $-15x² + 600x - 4{,}500 = 0$. Dividing the entire equation by $-15$ yields $x² - 40x + 300 = 0$. Factoring this quadratic equation gives $(x - 10)(x - 30) = 0$, so $x = 10$ or $x = 30$. The question asks for the smaller number of desks to break even, which is 10.

Choice B is incorrect because 20 is the number of desks that maximizes profit (the x-coordinate of the vertex of the parabola, $x = -600 / (2 \cdot (-15)) = 20$), not the break-even quantity.

Choice C is incorrect because 30 is the larger of the two break-even quantities.

Choice D is incorrect because 40 is the result of equating the unit price to the variable cost per desk ($900 - 15x = 300$), completely ignoring the fixed cost of $4,500$.