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The problem you're asking about has two parts, and I'll focus on the first one about the marginal profit.

### Given:
- Cost function: $ C(q) = 200 + 7q $
- Revenue function: $ R(q) = 17q $
- You want the **marginal profit** at $ q = 30 $.

### Step 1: Understand the marginal profit
The **marginal profit** is the rate of change of profit with respect to the number of items $ q $, i.e., $ \text{MP}(q) = R'(q) - C'(q) $. 

### Step 2: Derive the marginal cost and marginal revenue
1. The **marginal cost** is the derivative of the cost function, $ C'(q) $:
   \[
   C'(q) = \frac{d}{dq} \left( 200 + 7q \right) = 7
   \]
2. The **marginal revenue** is the derivative of the revenue function, $ R'(q) $:
   \[
   R'(q) = \frac{d}{dq} \left( 17q \right) = 17
   \]

### Step 3: Calculate the marginal profit
The marginal profit is:
\[
\text{MP}(q) = R'(q) - C'(q) = 17 - 7 = 10
\]
Thus, the marginal profit is **$10 per unit** when $ q = 30 $.

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Would you like more details or have further questions on this? Here are five related questions for deeper exploration:

1. What is the significance of marginal cost and marginal revenue in business decisions?
2. How is total profit related to marginal profit?
3. How would changes in fixed costs affect marginal cost?
4. What happens when marginal cost exceeds marginal revenue?
5. Can you explain how marginal profit is used to optimize production levels?

**Tip:** Marginal profit helps companies decide how many more units to produce by analyzing the additional profit from selling one more unit.

The cost function for a company to produce q items is C(q) = 200 + 7q. The revenue function for the same q items is R(q) = 17q. What is the marginal profit when q = 30?

Learn how to calculate the marginal profit when given a cost function C(q) = 200 + 7q and a revenue function R(q) = 17q. This problem involves using differentiation to find the marginal cost and marginal revenue, suitable for high school students in Grades 10-12.

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