If, in a monopoly market, the demand function for a product is 
p = 160 − 1.00x
 and the revenue function is 
R = px,
 where x is the number of units sold and p is the price per unit, what price will maximize revenue?

To find the price that maximizes revenue in a monopoly market, we need to follow these steps:

1. **Revenue function**:
   The revenue $ R $ is given by:
   \[
   R = px
   \]
   where $ p = 160 - 1.00x $ and $ x $ is the number of units sold.

2. **Substitute the demand function into the revenue function**:
   Substitute $ p = 160 - 1.00x $ into the revenue equation:
   \[
   R = (160 - 1.00x) \cdot x
   \]
   Simplifying the expression:
   \[
   R = 160x - 1.00x^2
   \]
   This is the revenue function in terms of $ x $.

3. **Differentiate the revenue function**:
   To maximize revenue, we need to take the derivative of $ R $ with respect to $ x $ and set it equal to zero:
   \[
   \frac{dR}{dx} = 160 - 2 \cdot 1.00x = 160 - 2x
   \]

4. **Find the critical point**:
   Set $ \frac{dR}{dx} = 0 $ to find the value of $ x $ that maximizes revenue:
   \[
   160 - 2x = 0
   \]
   Solving for $ x $:
   \[
   2x = 160 \quad \Rightarrow \quad x = 80
   \]

5. **Find the corresponding price**:
   Now that we know $ x = 80 $, substitute this value into the demand function $ p = 160 - 1.00x $ to find the price that maximizes revenue:
   \[
   p = 160 - 1.00(80) = 160 - 80 = 80
   \]

Thus, the price that maximizes revenue is **$80**.

Would you like further details, or have any questions?

Here are 5 related questions to explore this topic further:
1. How does the elasticity of demand affect the revenue-maximizing price?
2. What would happen to the revenue-maximizing price if the demand curve slope changes?
3. How can you find the profit-maximizing price if costs are included?
4. Can the revenue-maximizing price be higher or lower than the profit-maximizing price? Why?
5. How do you calculate marginal revenue from the demand function?

**Tip**: Remember that in a monopoly, the price set by the firm directly affects the quantity demanded and total revenue.

If, in a monopoly market, the demand function for a product is p = 160 − 1.00x and the revenue function is R = px, where x is the number of units sold and p is the price per unit, what price will maximize revenue?

Discover how to find the revenue-maximizing price in a monopoly market where the demand function is p = 160 - 1.00x and the revenue function is R = px. This detailed solution involves algebra and calculus, guiding you through each step, including finding critical points and solving for the optimal price.

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