Did the marketing group make a correct recommendation?

Question

The Wave Hotel offers luxury suites on Cocoa Beach. It has 200 beachfront suites. At the current price of $300 per night for each suite, it is fully occupied. A recent marketing study indicates that for every $10 increase in price, 5 fewer suites will be booked. (For example, if the price is $310, then 195 suites are booked. If the price is $320, then 190 suites are booked.)

The marketing group recommends that the Wave Hotel charge $300 per night for each suite to maximize the revenue.

 

Answer 1

(195, 310) (190, 320) are two points on a graph of (Q,P) or (x,y) with price measured on the y axis and number of suites rented on the x axis

line through the points has slope = (320-310)/(190-195) = 10/-5 = -2

(y-320)/(x-190) = -2

-2x + 380 = y-320

y = -2x +700, at a price of $700, no rooms are rented

at a price of $350, 175 rooms are sold and revenue = 175(350)= $61,250

at $300 revenue = 300(200) = $60,000, less than at a price of $350

revenue R = PQ = yx = -2x^2 + 700x

R' = -4x + 700 = 0

x = 175 = revenue maximizing quantity

y=-2(175)+700 = $350 = revenue maximizing price

R= xy= PQ = 175(350) = $61,250 = maximum revenue