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Given the following cost and revenue functions:

A.
Find the marginal cost
B.
Find the marginal revenue
C.
Find the marginal profit
D.
Find all values of x for which marginal profit is zero, and interpret.

We are given the cost function \( C(x) = 10x + 3 \) and the revenue function \( R(x) = 40x - 0.6x^2 \). Let's solve each part of the question.

### A. Find the Marginal Cost
The marginal cost is the derivative of the cost function \( C(x) \) with respect to \( x \).

\[
C'(x) = \frac{d}{dx}(10x + 3) = 10
\]

Thus, the marginal cost is \( 10 \).

### B. Find the Marginal Revenue
The marginal revenue is the derivative of the revenue function \( R(x) \) with respect to \( x \).

\[
R'(x) = \frac{d}{dx}(40x - 0.6x^2) = 40 - 1.2x
\]

Thus, the marginal revenue is \( 40 - 1.2x \).

### C. Find the Marginal Profit
The profit function \( P(x) \) is the difference between the revenue and the cost functions:

\[
P(x) = R(x) - C(x) = (40x - 0.6x^2) - (10x + 3) = 30x - 0.6x^2 - 3
\]

The marginal profit is the derivative of the profit function \( P(x) \) with respect to \( x \).

\[
P'(x) = \frac{d}{dx}(30x - 0.6x^2 - 3) = 30 - 1.2x
\]

Thus, the marginal profit is \( 30 - 1.2x \).

### D. Find all values of \( x \) for which the marginal profit is zero, and interpret
To find the values of \( x \) for which the marginal profit is zero, we set \( P'(x) = 0 \):

\[
30 - 1.2x = 0
\]

Solving for \( x \):

\[
1.2x = 30 \implies x = \frac{30}{1.2} = 25
\]

Thus, the marginal profit is zero when \( x = 25 \).

**Interpretation**: When the production level is 25 units, the marginal profit becomes zero, which means that producing an additional unit beyond this point will neither increase nor decrease profit. This is typically the point where profit is maximized.

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Would you like further details or have any questions?

Here are 5 related questions:
1. How is marginal cost related to total cost?
2. What does it mean if the marginal revenue is less than the marginal cost?
3. How can we find the maximum profit using marginal functions?
4. What happens to profit if the marginal profit is negative?
5. What role do derivatives play in determining marginal values?

**Tip**: Marginal cost and marginal revenue help businesses decide the optimal production level. Maximizing profit occurs when marginal revenue equals marginal cost!

Given the following cost and revenue functions:

A. Find the marginal cost
B. Find the marginal revenue
C. Find the marginal profit
D. Find all values of x for which marginal profit is zero, and interpret.

Learn how to determine the marginal cost, marginal revenue, and marginal profit for the given cost function C(x) = 10x + 3 and revenue function R(x) = 40x - 0.6x^2. This problem also involves finding the production level where the marginal profit is zero. Suitable for high school students in Grades 10-12.

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