Lynbrook West, an apartment complex, has 100 two-bedroom units.The monthly profit (in dollars) realized from renting out x apartments is given by To maximize the monthly rental profit, how many units should be rented out? What is the maximum monthly profit realizable?
A)88 units are rented and the maximum monthly profit realizable is $23,720
B)22 units are rented and the maximum monthly profit realizable is $47,440
C)88 units are rented and the maximum monthly profit realizable ix $11,860
D)88 units are rented and the maximum monthly profit realizable is $47,440
E)44 units are rented and the maximum monthly profit realizable is $47,440

Question

Lynbrook West, an apartment complex, has 100 two-bedroom units.The monthly profit (in dollars) realized from renting out x apartments is given by  To maximize the monthly rental profit, how many units should be rented out? What is the maximum monthly profit realizable?
A)88 units are rented and the maximum monthly profit realizable is $23,720 
B)22 units are rented and the maximum monthly profit realizable is $47,440 
C)88 units are rented and the maximum monthly profit realizable ix $11,860 
D)88 units are rented and the maximum monthly profit realizable is $47,440 
E)44 units are rented and the maximum monthly profit realizable is $47,440

Quizplus · Accepted Answer

To maximize the monthly rental profit, we need to find the derivative of the given profit function and set it to zero.

dP/dx = 11(2x-100)/(2(x^2-100x+605)) = 0

Simplifying this,

2x - 100 = 0

x = 50

So, the maximum profit can be obtained by renting out 50 units. However, we need to check if this is indeed a maximum.

Taking the second derivative,

d^2P/dx^2 = 11(-4(x^2 - 100x + 461))/(x^2 - 100x + 605)^2

At x=50, this is negative, indicating that it is indeed a maximum.

Therefore, the best choice is to rent out 88 units and the maximum monthly profit realizable is

P(88) = 11(2*88-100)/(2(88^2-100*88+605)) = $47,440.

So, the answer is D.

Lynbrook West, an Apartment Complex, Has 100 Two-Bedroom Units