𝑞 𝑝, 𝑀 = 0.1𝑀 − 50𝑝
Let M=10,000. Let us calculate the following:
1. The total revenue function.
2. The average revenue function.
3. The marginal revenue function.

Given the equation \( q(p, M) = 0.1M - 50p \), where \( M = 10,000 \), let's proceed to solve the three requested items:

### 1. **The Total Revenue Function**

Revenue, \( R(p) \), is given by the product of price \( p \) and quantity \( q \). From the given equation, we know that \( q = 0.1M - 50p \). Substituting \( M = 10,000 \):

\[
q(p) = 0.1(10,000) - 50p = 1,000 - 50p
\]

The total revenue function is:

\[
R(p) = p \cdot q(p) = p \cdot (1,000 - 50p)
\]

Thus, the total revenue function is:

\[
R(p) = 1,000p - 50p^2
\]

---

### 2. **The Average Revenue Function**

The average revenue function \( AR(p) \) is the total revenue divided by the quantity, i.e., \( AR(p) = \frac{R(p)}{q(p)} \).

We already found the total revenue function \( R(p) = 1,000p - 50p^2 \) and the quantity function \( q(p) = 1,000 - 50p \).

Thus, the average revenue function is:

\[
AR(p) = \frac{1,000p - 50p^2}{1,000 - 50p}
\]

This is the average revenue function.

---

### 3. **The Marginal Revenue Function**

The marginal revenue function \( MR(p) \) is the derivative of the total revenue function with respect to \( p \).

\[
R(p) = 1,000p - 50p^2
\]

Now, differentiate this function with respect to \( p \):

\[
MR(p) = \frac{d}{dp} \left(1,000p - 50p^2\right) = 1,000 - 100p
\]

Thus, the marginal revenue function is:

\[
MR(p) = 1,000 - 100p
\]

---

### Summary:

1. **Total Revenue Function**: \( R(p) = 1,000p - 50p^2 \)
2. **Average Revenue Function**: \( AR(p) = \frac{1,000p - 50p^2}{1,000 - 50p} \)
3. **Marginal Revenue Function**: \( MR(p) = 1,000 - 100p \)

Would you like more details on any of these results or have any questions?

### 5 Related Questions:
1. What does the total revenue function represent in practical terms?
2. How does the price \( p \) affect the marginal revenue?
3. What happens to the total revenue as \( p \) increases or decreases?
4. How do we interpret a negative marginal revenue in this context?
5. How would the functions change if \( M \) were different?

### Tip: 
The marginal revenue tells you how much the total revenue changes when you sell one more unit. Keep an eye on when it becomes negative—it signals diminishing returns from increasing price.

The query submitted by the user: q(p, M) = 0.1M - 50p. Let M = 10,000. Let us calculate the following: 1. The total revenue function. 2. The average revenue function. 3. The marginal revenue function.

Learn how to calculate the total revenue, average revenue, and marginal revenue using a given demand function q(p, M) = 0.1M - 50p, where M = 10,000. Explore step-by-step solutions involving algebra and differentiation to analyze key revenue functions. Suitable for undergraduate students.

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