rate$ #of rooms, & max revenue?

Question

if 52 rooms sell at a price of $90
for every $5 increase 4 fewer peop;e buy a room
there is a max of 56 rooms to sell

at what rate should the rooms sell at?
at that price how many rooms will sell?
&
What is thae max revenue?

Thanks Just need to check the answer I got

Answer 1

52 rooms sell at $90 each yielding $4,680
56 rooms available
n - 52 = -(4/5)(r - 90)
n = 124 - (4/5)r, r ≥ 85

56*$85 = $4,760

55*$86.25 = $4,743.75
54*$87.50 = $4,725
52*$90 = $4,680
48*$95 = $4,560

Answer 2

f(x) = (52-4n)(90+5n) (where n is the number of people)
f(x) = 4680-20n^2-100n
f'(x)= -40n-100

Solve for 0

-40n-100= 0
n= -2.5
.'. n = -3
,', The rooms should be sold for $75
The max revenue = 64*75 = $4736

That is probably wrong but eh

If its for a test u should probably varify max/min by finding the second derivitive i.e d^2x/dx^2

In this case d^2x/dx^2 = -40 .'. max ( >1 = min <1 = max)

Answer 3

56 rooms
$85/room
$4760

Answer 4

90$ total or each ?