We are given the profit function $P(q) = 1200 - 10q - q^2$, where $q$ is the quantity of units produced and sold. We want to find the number of units that must be produced and sold to break even, meaning the profit is zero, $P(q) = 0$. Thus we need to solve the equation $1200 - 10q - q^2 = 0$ for $q$.

1. Problem Description

We are given the profit function $P(q) = 1200 - 10q - q^2$, where $q$ is the quantity of units produced and sold. We want to find the number of units that must be produced and sold to break even, meaning the profit is zero, $P(q) = 0$. Thus we need to solve the equation $1200 - 10q - q^2 = 0$ for $q$.

2. Solution Steps

We need to solve the quadratic equation

$1200 - 10q - q^2 = 0$

Rearranging the terms, we have:

$q^2 + 10q - 1200 = 0$

We can factor the quadratic expression:

$(q - 30)(q + 40) = 0$

So, $q - 30 = 0$ or $q + 40 = 0$.

This gives us two possible values for $q$:

$q = 30$ or $q = -40$.

Since $q$ represents the number of units produced and sold, it must be a non-negative number. Therefore, $q = -40$ is not a valid solution. Thus, $q = 30$.

3. Final Answer

The number of units that must be produced and sold to break even is

3

0. Answer: (b)