calculus question

Question

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $500 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 Increase in rent. What rent should the manager charge to maximize revenue?

Answer 1

A different approach, same answer.

Revenue, R(x) = Quantity × Price

Where x is a multiplier.

Quantity = 100-1x

Price = 500+10x

R(x) = (100-x)(500+10x) = 50000+500x-10x²

To maximize, find R'(x), set that to zero and solve for x.

R'(x) = -20x+500

-20x = -500

x = 500/20 = 25

Rent = 500+25(10) = $750

Revenue = R(25)= 75(750) = $56,250

Which is $6,250 more than not raising the rent.

Answer 2

100 x 500, = 50,000

99 x 510 = 50,490

rxp=R= rooms times price=Revenue

(100, 500), (99, 510)=(r,p)

slope=-10= (p-500)/(r-100)

p= -10r+1500

R=pr= -10r^2+1500r

R'(r) =-20r +1500=0

r=1500/20= 75

p=-750+1500

= $750= R maximizing price

max R=75x750= $56,250

760x74=56240

740x76=56240

only question is what happens if they raise the rental price by $9 to $759? does the extra vacancy occur when it's only a $10 jump? if so, then charge $759 to get $759 x 75 = $56,925 = max R

or raise it $9.99 to $759.99 x 75 = $56,999.25= max R