Quadratic Functions

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.

Did the marketing group make a correct recommendation? Hint: Revenue is the total amount of money brought in.

Answer 1

(200, 300), (195, 310) are two points on the demand curve

it has slope = -10/5 = -2

P= -2Q + b, solve for b by plugging in either point

300=-2(200) + b

b = 700

Revenue = PQ = -2Q^2 + 700Q

take the derivative of Revenue and set it equal to zero. Solve for Q, then plug it into the demand function to find the revenue maximizing price

R' = -4Q + 700 = 0

Q = 700/4 = 175

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

you rent 25 less rooms, losing 25x300= 7500

but you gain an extra $50 per room for plus 50x175= 8750 or a net gain in revenue of 1250 over a $300 price