To find the positive output level that maximizes the firm's profit, we need to determine the derivative of the profit function and set it equal to zero. Let's calculate the derivative:
\[
\pi'(q) = \frac{d}{dq} (120q - (300 + 20q + 10q^2))
\]
Simplifying the derivative:
\[
\pi'(q) = 120 - 20 - 20q
\]
Setting the derivative equal to zero:
\[
120 - 20 - 20q = 0
\]
Simplifying the equation:
\[
100 - 20q = 0
\]
Solving for q:
\[
20q = 100 \quad \Rightarrow \quad q = \frac{100}{20} \quad \Rightarrow \quad q = 5
\]
Therefore, the positive output level that maximizes the firm's profit is \( q = 5 \).
To find the firm's profit at this output level, we substitute \( q = 5 \) into the profit function:
\[
\pi(5) = 120(5) - (300 + 20(5) + 10(5^2))
\]
Simplifying:
\[
\pi(5) = 600 - (300 + 100 + 250)
\]
\[
\pi(5) = 600 - 650
\]
\[
\pi(5) = -50
\]
Therefore, the firm's profit at the output level of 5 units is \(-50\), indicating a loss of $50.