Example
Zoe is the only lemonade seller on her street. People who drive home stop by to purchase liters of her lemonade.
The demand for lemonade is:
$$ P = 10 - 2Q $$When Zoe produces a quantity `Q` of lemonade, her revenue is
$$ R \left( Q \right) = PQ = \left( 10 - 2Q \right) Q $$Her cost follows the equation
$$ C \left( Q \right) = 2 Q $$Overall, her profit is
$$ \Pi \left( Q \right) = R \left( Q \right) - C \left( Q \right) = \left( 10 - 2Q \right) Q - 2 Q = 8Q - 2 Q^2 $$If Zoe makes 2 liters of lemonade (Q = 2), her profit is
$$ \Pi \left( 2 \right) = 8 \times 2 - 2 \times 2^2 = 8 $$She would make a profit of $8.
Question
The inverse demand is `P = 51 - 9Q` and the costs of production are `18 Q`.
What is Zoe's profit is she produces `Q=4`L of lemonade.
The revenue is
$$ R \left( Q \right) = PQ = (51 - 9Q) Q = 51 Q - 9Q^2 $$The cost is
$$ C ( Q ) = 18 Q $$So the profit function is
$$ \begin{align*} \Pi ( Q ) &= R (Q ) - C ( Q) \\ &= 51 Q - 9Q^2 - 18 Q \\ &= 33 Q - 9 Q^2 \end{align*} $$When `Q=4`
$$ \Pi ( 4 ) = 33 \times 4 - 9 \times 4^2 = -12 $$