To derive the demand function $q(p)$ from the given revenue function, we first need to understand the relationship between revenue, price, and quantity.

The revenue function $R(p)$ is typically expressed as: $R(p) = p \cdot q(p)$ where $p$ is the price, and $q(p)$ is the demand function (quantity demanded at price $p$).

We are given the revenue function: $R(p) = -53p^2 + 69p$ Using the equation $R(p) = p \cdot q(p)$, we can solve for $q(p)$ by dividing the revenue function by $p$: $q(p) = \frac{R(p)}{p} = \frac{-53p^2 + 69p}{p}$

Simplifying: $q(p) = -53p + 69$

So, the demand function is: $q(p) = -53p + 69$

Reason for the answer:

We found the demand function by recognizing that revenue is the product of price and quantity ($R = p \cdot q$). By dividing the revenue function by price, we isolated the demand function.

Expanded questions:

1. How does the demand function affect the price as $p$ increases?
2. What happens to the quantity demanded as price approaches 0?
3. What is the slope of the demand function and what does it represent economically?
4. Can you find the price where revenue is maximized?
5. What is the significance of the intercept in the demand function?

Tip:

Always remember that revenue is the product of price and quantity when working with these types of functions.