Help me solve this pls

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?

Answer 1

Revenue = PQ = Price times quantity

300x200 = $60,000 for $300 per room, 200 rooms

310x 195 = $60,450

320 x 190= $60,800

330 x185= $61,500

340 x 180= $61,200

350 x 175 = $61,250

360x170= $61,200

$350 brings in $1,250 more profits than $300. marketing group got it wrong, but still close, not that far off

R=PQ = P(200-.5(P-300)) = P(200-.5P+150) = 350P -.5P^2

R' = 350-P=0

P=$350 = profit maximizing price

either below or above 350 brings less revenue

(this unrealistically assumes costs are zero or unaffected by number of rooms rented)

write the Revenue equation solely in terms of one variable, either Price or Quantity

then take the derivative, set = 0 and solve for the variable.

Q = 350 -P/2

P/2 = 350-Q

P = 700-2Q

R = PQ = (700-2Q)Q = 700Q -2Q^2

R' = 700 -4Q = 0

Q =700/4= 175

P= 700-2Q = 700-2(175)= 700-350= $350