Let's solve each part step by step:

Part a: Expression for Marginal Revenue

The marginal revenue is the derivative of the revenue function $R(x)$ with respect to $x$. Given the revenue function:

$R(x) = 500 - x^2$

we differentiate $R(x)$ with respect to $x$:

$R'(x) = \frac{d}{dx}(500 - x^2) = -2x$

Thus, the expression for the marginal revenue is:

$R'(x) = -2x$

Part b: Marginal Revenue for 50 Units Sold

To find the marginal revenue when 50 units are sold, we substitute $x = 50$ into $R'(x)$:

$R'(50) = -2(50) = -100$

The marginal revenue when 50 units are sold is $-100$ dollars. This means that if the number of units sold increases from 50 to 51, the revenue is expected to decrease by 100 dollars.

Part c: Marginal Revenue for 250 Units Sold

Similarly, substitute $x = 250$ into $R'(x)$:

$R'(250) = -2(250) = -500$

The marginal revenue when 250 units are sold is $-500$ dollars. This indicates that if the number of units sold increases from 250 to 251, the revenue is expected to decrease by 500 dollars.

Part d: Analysis of Revenue as Units Pass Through 250

Since the marginal revenue is negative at both 50 units ($R'(50) = -100$) and 250 units ($R'(250) = -500$), the revenue is decreasing at these points. As $R'(x) = -2x$ becomes more negative as $x$ increases, the rate of revenue decrease is accelerating. Therefore, as the number of units sold passes through 250, the revenue is not changing from decreasing to increasing; instead, the rate at which it is decreasing is becoming faster.

Summary:

• The marginal revenue function is $R'(x) = -2x$.
• At 50 units, marginal revenue is $-100$, indicating revenue decreases by $100 for each additional unit sold.
• At 250 units, marginal revenue is $-500$, meaning revenue decreases by $500 for each additional unit sold.
• As units sold pass through 250, the revenue continues to decrease, but at an accelerating rate.

Would you like further details or have any questions?

Related Questions:

1. What is the interpretation of a positive marginal revenue?
2. How can the revenue function be maximized for this sound system?
3. What would the revenue be at zero units sold?
4. How does changing the revenue function affect marginal revenue?
5. What is the significance of the second derivative in analyzing the revenue function?

Tip:

Remember that the marginal revenue indicates the rate of change of the revenue; when it is negative, revenue decreases as the number of units sold increases.