Question

An apartment complex has 250 departments to rent. If they rent x apartments then their monthly profit, in dollars, is given by

$P\left(x\right)=-8x^2+3200x-80000$

How many apartments should they rent in order to maximize the profit ?

Answer 1

Here, we have to maximize the profit subject to the constraint that x must be in range $0\le x\le 250$.

Now,

$P^{\prime }\left(x\right)=-16x+3200$

Putting $P^{\prime }\left(x\right)=0$

$16x=3200$

$x=200$

Finding the value of P(x) at $x=200$ and at the end points,

$P\left(0\right)=-80000$

$P\left(200\right)=240000$

$P\left(250\right)=220000$

So, they will generate the most profit if they only rent out 200 of the apartments instead of all 250 of them.