Answer to Question #338108 in Calculus for juju

Question

All 77 rooms in a motel will be rented each night if the manager charges $40 or less per room. If he charges $(40+z) per room, then 2z rooms will remain vacant. If each rented room costs the manager $7 per day and each unrented room $2 per day in overhead, how much should the manager charge per room to maximise his profit?

Answer 1

If 2z rooms are vacant, then 77-2z are rented. For all rented rooms he will get (77-2z)(40+z)$, Rented room will costs for him 7(77-2z)$, and not rented rooms will cost for him 2x2z=4z$.

So profit is P=(77-2z)(40+z)-7(77-2z)-4z=(77-2z)(40+z-7)-4z=(77-2z)(33+z)-4z=2541+77z-66z-2z²-4z=2541+7z-2z²

$P'=7-4z=0$

z=7/4

$P''=-4<0$ , hence z=7/4 is a point of maximum.

Answer: 7/4$.