## Example

Zoe, in her lemonade stand, faces the inverse demand `P = 10 - 2Q`. So her revenue is

$$ R ( Q ) = PQ = (10 - 2Q)Q = 10 Q - 2Q^2 $$Her marginal revenue is

$$ MR ( Q ) = \frac{d R ( Q )}{d Q} = 10 - 4 Q $$Zoe faces production costs equal to `C ( Q ) = 2Q`. So her marginal costs are

$$ MC ( Q ) = \frac{d C ( Q )}{d Q} = 2 $$The quantity Q that maximizes Zoe's profit solves

$$ \begin{align*} MC ( Q ) &= MR ( Q ) \\ 2 &= 10 - 4 Q \\ 4 Q &= 10 - 2 \\ 4 Q &= 8 \\ Q &= 2 \end{align*} $$### Question

Zoe's satisfies an inverse demand `P = 331 - 3Q`.

Her production costs are `13 Q+ 91`.

What quantity of lemonade should Zoe produce to maximize profit?

Zoe maximizes profit when producing `Q=53`L of lemonade.

With a cost function `13 Q+ 91`, Zoe's marginal cost is

$$ MC \left( Q \right) = \frac{d C ( Q )}{d Q} = 13 $$With an inverse demand `P = 331 - 3Q`, Zoe's revenue is

$$ R \left( Q \right) = (331 - 3Q) Q = 331 Q - 3Q^2 $$Her marginal revenue is

$$ MR \left( Q \right) = \frac{d R ( Q )}{d Q} = 331 - 3 \times 2 Q = 331 - 6Q $$The profit-maximizing quantity satisfies

$$ \begin{align*} MC \left( Q \right) &= MR \left( Q \right) \\ 13 &= 331 - 6Q \\ 6 Q &= 331 - 13 \\ 6 Q &= 318 \\ Q &= 53 \\ \end{align*} $$