Optimal number of light bulbs for maximum profit

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The company Meucciolight produces light bulbs. The production cost is $10$ thalers each. If $1000$ are put on the market the selling price turns out to be $20$ thalers; if $500$ are put out the price is $30$ thalers. Find the optimal number of bulbs to produce to maximize profit, compute that maximum profit and find the selling price at the optimum.

Solution

Linear demand curve $q=mp+b$. From the points $(p,q)=(20,1000)$ and $(30,500)$: $m=\frac{500-1000}{30-20}=-50,\qquad 1000=-50\cdot 20+b\Rightarrow b=2000\quad\Rightarrow\quad q=2000-50p.$ Profit: $\pi(p)=(p-10)(2000-50p)=-50p^2+2500p-20000.$ Vertex (maximum): $p^\ast=-\frac{2500}{2\cdot(-50)}=25\ \text{thalers}.$ Quantity $q=2000-50\cdot 25=750$ bulbs; profit $\pi_{\max}=(25-10)\cdot 750=11250\ \text{thalers}.$

$\boxed{q^\ast=750,\quad p^\ast=25\ \text{thalers},\quad \pi_{\max}=11250\ \text{thalers}}$