Let’s break down the problem part by part.

(a) Interpretation of $f(20) = 15,000$ and $f'(20) = -300$:

• $f(p)$ represents the quantity of fabric sold as a function of the price per yard, $p$.

• $f(20) = 15,000$ means that when the price of the fabric is $20 per yard, the manufacturer sells 15,000 yards of fabric.

• $f'(20) = -300$ represents the rate of change of the quantity sold with respect to the price at $p = 20$. This means that when the price is $20 per yard, an increase of $1 in price will result in a decrease in the quantity sold by 300 yards. In other words, the demand is decreasing as the price increases.

(b) Finding $R'(20)$ and its interpretation:

The total revenue function $R(p)$ is given by: $R(p) = p \cdot f(p)$ To find the rate of change of revenue with respect to price, $R'(p)$, we use the product rule for differentiation: $R'(p) = f(p) + p \cdot f'(p)$ Substitute the given values $f(20) = 15,000$, $f'(20) = -300$, and $p = 20$: $R'(20) = f(20) + 20 \cdot f'(20)$ $R'(20) = 15,000 + 20 \cdot (-300)$ $R'(20) = 15,000 - 6,000 = 9,000$

Interpretation of $R'(20) = 9,000$:

This means that when the price is $20 per yard, the revenue is increasing at a rate of $9,000 per dollar increase in price. Thus, increasing the price by $1 from $20 will result in a $9,000 increase in total revenue.

Summary:

• $f(20) = 15,000$ means 15,000 yards are sold when the price is $20.
• $f'(20) = -300$ means that for every $1 increase in price, 300 fewer yards are sold.
• $R'(20) = 9,000$ means that the revenue is increasing by $9,000 for each $1 increase in price when the price is $20.

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Would you like more details or have any further questions? Here are five related questions for deeper understanding:

1. How would the total revenue behave if $f'(p)$ were positive instead of negative?
2. How is elasticity of demand related to the values of $f'(p)$?
3. What would happen to $R'(p)$ if $f'(p)$ were smaller in magnitude?
4. Can you determine the price that maximizes revenue based on this information?
5. How does the interpretation of $f'(p)$ change if we measure price in different units?

Tip: The product rule is crucial when dealing with the derivative of functions involving both price and quantity in revenue or cost models.