https://file.aiquickdraw.com/mathbot/file/17267797260601akf18ix.jpg

The problem in the image involves finding the **marginal revenue** for a given demand equation. Let's work through it step by step.

The **demand equation** is given as:

\[
x = 7000 - 100p
\]

Where:
- $ x $ is the quantity demanded,
- $ p $ is the price.

### Step 1: Solving for price $ p $
We need to express $ p $ in terms of $ x $:

\[
x = 7000 - 100p
\]

Solve for $ p $:

\[
100p = 7000 - x
\]

\[
p = \frac{7000 - x}{100}
\]

### Step 2: Revenue Function
Revenue $ R(x) $ is the product of the quantity demanded $ x $ and the price $ p $:

\[
R(x) = x \cdot p = x \left( \frac{7000 - x}{100} \right)
\]

Simplify this:

\[
R(x) = \frac{x(7000 - x)}{100} = \frac{7000x - x^2}{100}
\]

### Step 3: Marginal Revenue
Marginal revenue is the derivative of the revenue function with respect to $ x $:

\[
R'(x) = \frac{d}{dx} \left( \frac{7000x - x^2}{100} \right)
\]

Differentiate the expression:

\[
R'(x) = \frac{7000 - 2x}{100}
\]

This is the **marginal revenue function**.

### Step 4: Calculate Marginal Revenue for $ x = 1500 $
Substitute $ x = 1500 $ into the marginal revenue function $ R'(x) = \frac{7000 - 2x}{100} $:

\[
R'(1500) = \frac{7000 - 2(1500)}{100} = \frac{7000 - 3000}{100} = \frac{4000}{100} = 40
\]

Thus, the marginal revenue at $ x = 1500 $ units is **$40 per unit**.

### Step 5: Answer Selection
The correct answer corresponds to option **A**, which states:

> The marginal revenue at 1500 units is $40 per unit.

Let me know if you need further explanation or additional details!

---

### Related Questions:
1. How do you calculate marginal revenue when given the demand equation?
2. What is the significance of marginal revenue in economics?
3. How would you find the price at which demand equals 2000 units in this problem?
4. What happens to marginal revenue as $ x $ increases?
5. How is total revenue different from marginal revenue?

### Tip:
When solving for marginal revenue, always remember to first express revenue in terms of the demand variable $ x $ before differentiating.

Assume that a demand equation is given by x = 7000 - 100p. Find the marginal revenue for the given production levels (values of x). What is the marginal revenue function, R'(x)?

Learn how to find the marginal revenue from a demand equation given by x = 7000 - 100p. This example covers key mathematical concepts such as revenue functions, marginal revenue, and using derivatives to solve for the rate of change in revenue. Suitable for high school and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation