Questions: Find the revenue and demand functions for the given marginal revenue. (Use the fact that R=0 when x=0.) dR/dx=390-18x revenue function R= demand function p=

Solution Steps

To find the revenue function \( R(x) \), we need to integrate the given marginal revenue function \(\frac{dR}{dx} = 390 - 18x\). The constant of integration can be determined using the condition \( R(0) = 0 \). Once we have the revenue function, the demand function \( p(x) \) can be found by dividing the revenue function by \( x \), since \( R(x) = p(x) \cdot x \).

To find the revenue function \( R(x) \), we integrate the marginal revenue function:

\[ \frac{dR}{dx} = 390 - 18x \]

Integrating gives:

\[ R(x) = -9x^2 + 390x + C \]

Using the condition \( R(0) = 0 \), we find \( C = 0 \). Thus, the revenue function simplifies to:

\[ R(x) = -9x^2 + 390x \]

The demand function \( p(x) \) can be derived from the revenue function using the relationship \( R(x) = p(x) \cdot x \). Therefore, we have:

\[ p(x) = \frac{R(x)}{x} = \frac{-9x^2 + 390x}{x} = -9x + 390 \]

Final Answer