## Example

Zoe's lemonade stand satisfies a demand characterized by:

$$ P = 10 - 2 Q $$Her revenue is

$$ R \left( Q \right) = PQ = \left( 10 - 2 Q \right) Q = 10 Q - 2 Q^2 $$The marginal revenue is the derivative of the revenue `R \left( Q \right)`:

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 10 - 2 \times 2 Q = 10 - 4 Q $$### Question

The inverse demand for lemonade is `P = 85 - 5Q`.

What is Zoe's marginal revenue?

First, her revenue is

$$ R \left( Q \right) = PQ = (85 - 5Q) Q = 85 Q - 5Q^2 $$So her marginal revenue is

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 85 - 5 \times 2 Q = 85 - 10Q $$