Answer to Question #223578 in Calculus for Ayoze

Question

all 77 rooms in a motel will be rented each night if the maanager charges $39 or less per room. if he charges $(39 +y) per room then 2y rooms will remain vacant. If each rented room costs the manager $5 per day and each unrented room $3 per day in overhead , how much should the manager charge per room to maximize his profit. find;

i) no of rented rooms

ii)the revenue

iii)total cost in overhead

Answer 1

i)  no of rented rooms

$$77-2y, 0\leq y\leq38$$

ii) the revenue

$$R=(39+y)(77-2y)=-2y^2-y+3003$$

iii) total cost in overhead

$$C=5(77-2y)+3(2y)=385-4y$$

Profit

$$P=R-C=-2y^2-y+3003-(385-4y)$$

$$P=P(y)=-2y^2+3y+2618, 0\leq y\leq38$$

$$P'(y)=-4y+3$$

$$P'(y)=0=>-4y+3=0=>y=\dfrac{3}{4}$$

$$P(0)=2618$$

$$P(1)=-2+3+2618=2619$$

$$39+1=40$$

The manager should charge $40 per room to maximize his profit.