What rent should the manager charge to maximize revenue?

Question

The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 396 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 9 dollar increase in rent. Similarly, one additional unit will be occupied for each 9 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

Okay, let's take t as the number of nine dollar increments we'll raise the rent, R as the price of Rent, and N as the number of apartments that will be rented.

 

we know that the number of apartments that will rent is 100 minus t, and we know that the price is 396 plus 9*t, and the total income is the product of those two numbers

 

N(t) = (100 - t)

R(t) = (396+9t)

 

$(t) = (100 - t)(396+9t)

$(t) = 39600 +900t - 396t -9t^2

$(t) = -9t^2 + 504t + 39600

 

Depending on what kind of class you are in, you could either use the quadratic equation to find the two zeros of this equation and the best rent will be halfway between, or you could take the derivative and find the zero of that.  

 

It's calculus, so we'll do the derivative.

 

$'(t) = -18t + 504

 

t = 504/18 = 28

 

so the proper rent is 

 

R(28) = (396+9*28) = 648