1. **State the problem:** We are given the marginal cost function $C'(q) = 500 + 3q + q^2$ and asked to find the value of $C'(5)$, which represents the marginal cost when producing 5 units. 2. **Recall the formula:** The marginal cost function $C'(q)$ gives the rate of change of the total cost with respect to the number of units $q$. To find $C'(5)$, we substitute $q=5$ into the function. 3. **Substitute and calculate:** $$C'(5) = 500 + 3(5) + (5)^2$$ 4. **Simplify step-by-step:** $$C'(5) = 500 + 15 + 25$$ 5. **Add the terms:** $$C'(5) = 540$$ 6. **Interpretation:** The marginal cost of producing the 5th unit is 540 dollars. **Final answer:** $$\boxed{540}$$