Optimal profit price for luxury cars

Text

The Fantaserati car company sells luxury cars. At a price of $100\,000$ € per car it sells $300$ cars per year; at a price of $50\,000$ € per car it sells $1000$ per year. With a production cost of $30\,000$ € per car, determine the price that yields optimal profit (revenue minus cost).

Solution

Assume the number of cars sold $q$ depends linearly on the price $p$. The line passes through $(100\,000;\,300)$ and $(50\,000;\,1000)$: $m=\frac{1000-300}{50\,000-100\,000}=\frac{700}{-50\,000}=-0{,}014,\qquad q=1700-0{,}014\,p.$ The profit is $G=(\text{price}-\text{cost})\cdot q$: $G(p)=(p-30\,000)(1700-0{,}014\,p)=-0{,}014\,p^2+2120\,p-51\,000\,000.$ This is a downward parabola: the maximum is at the vertex $p=-\frac{2120}{2\cdot(-0{,}014)}=\frac{2120}{0{,}028}=\frac{530\,000}{7}\approx 75\,714{,}29.$ At that price $q=1700-0{,}014\cdot75\,714\approx 640$ cars are sold, for a profit of about $45\,714\cdot640\approx 29.3$ million euros. $\boxed{p\approx 75\,714\ \text{€ per car.}}$