Please Help ASAP

Question

A 320-unit hotel is trying to determine their daily rates. From experience, the hotel manager knows that all rooms are typically occupied when the daily room rate is $89. Market research indicates that for every increase of x dollars in the daily room rate, there will be x rooms vacant. Each occupied room costs $22 per day to service and maintain. What should the hotel charge per day in order to maximize profit?

Answer 1

$89 per day, for 320 room capacity

increase price by $1, 319 rooms are rented

for every one dollar increase one room less is rented

(320, 89) and (319,90) have a line through them with slope = -1

P =-(x-320) + 89

= -x + 320+89

= -x + 409

Revenue = Px= -x^2 + 409x

R' = -2x +409

Cost = C = 22x

C' = 22

R'-C' = -2x +409 -22

= -2x + 382

x = 382/2 = 191= revenue maximizing rooms rented

P = -(191-320) +89

= -(-129) +89

= 129 + 89

= 218

= $218 per room maximizes profits