If the revenue function for a particular commodity is
R(p) = −57p2 + 66p,
what is the (linear) demand function q(p)?
q(p) =

Give a reason for your answer.

R = pq

R =
q
p

R = q − p

R =
p
q

R = p + q

The revenue function \( R(p) \) for a commodity is given by:

\[
R(p) = -57p^2 + 66p
\]

We know that revenue is the product of price \( p \) and quantity demanded \( q(p) \):

\[
R(p) = p \cdot q(p)
\]

Thus, the demand function \( q(p) \) can be expressed as:

\[
q(p) = \frac{R(p)}{p}
\]

Substitute \( R(p) = -57p^2 + 66p \) into the equation:

\[
q(p) = \frac{-57p^2 + 66p}{p}
\]

Simplifying the right-hand side:

\[
q(p) = -57p + 66
\]

Thus, the demand function is:

\[
q(p) = -57p + 66
\]

### Reason:
Revenue is calculated as \( R(p) = p \cdot q(p) \), meaning the demand function is obtained by dividing the revenue function by \( p \). This results in a linear demand function in terms of \( p \).

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How is marginal revenue derived from the revenue function?
2. What happens to demand as the price \( p \) increases, according to this demand function?
3. How would you find the price elasticity of demand from this function?
4. What is the significance of the slope in the demand function \( q(p) = -57p + 66 \)?
5. How would the revenue function change if demand were elastic?

**Tip:** The slope of the demand function represents the rate of change in quantity demanded with respect to price changes.

If the revenue function for a particular commodity is R(p) = −57p^2 + 66p, what is the (linear) demand function q(p)?

This guide explains how to derive the linear demand function from the given revenue function R(p) = −57p^2 + 66p. Learn the step-by-step process, including the formula q(p) = R(p) / p and concepts in algebra and revenue analysis. Ideal for college or advanced high school students.

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