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.