To solve the problem, let's break it down into two parts:

(i) Finding the Revenue Function $R(x)$

The marginal revenue function $R'(x)$ is given by:

$R'(x) = 15 - 9x - 3x^2$

The revenue function $R(x)$ is the integral of the marginal revenue function. We can find it by integrating $R'(x)$ with respect to $x$:

$R(x) = \int (15 - 9x - 3x^2) \, dx$

Let's integrate each term:

$\int 15 \, dx = 15x$

$\int -9x \, dx = -\frac{9x^2}{2}$

$\int -3x^2 \, dx = -\frac{3x^3}{3} = -x^3$

So, the revenue function is:

$R(x) = 15x - \frac{9x^2}{2} - x^3 + C$

Where $C$ is the constant of integration, which can be determined if initial conditions are provided (e.g., if the revenue is known for a certain quantity $x$).

(ii) Finding the Demand Function $p(x)$

The demand function $p(x)$ relates the price $p$ to the quantity $x$. In economics, revenue $R(x)$ is often given by:

$R(x) = p(x) \cdot x$

To find the demand function, express $p(x)$ as:

$p(x) = \frac{R(x)}{x}$

Substituting the revenue function $R(x)$ we found earlier:

$p(x) = \frac{15x - \frac{9x^2}{2} - x^3 + C}{x}$

Simplifying:

$p(x) = 15 - \frac{9x}{2} - x^2 + \frac{C}{x}$

So, the demand function is:

$p(x) = 15 - \frac{9x}{2} - x^2 + \frac{C}{x}$

Summary

1. Revenue Function: $R(x) = 15x - \frac{9x^2}{2} - x^3 + C$
2. Demand Function: $p(x) = 15 - \frac{9x}{2} - x^2 + \frac{C}{x}$

If you want to find the specific value of $C$, you'll need additional information about revenue at a particular quantity.

Would you like more details or have any further questions?

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Here are some questions to explore this topic further:

1. How does the value of $C$ affect the revenue and demand functions?
2. What is the significance of the marginal revenue function in determining optimal production levels?
3. How can the demand function help in pricing strategy?
4. What happens to the revenue function when $x$ increases significantly?
5. How does the shape of the demand curve change with different values of $C$?

Tip: Always check initial conditions or specific data points to determine constants when integrating functions in economics.