I want to increase my rent of $800 On the 500 apt units I own. I think for every increase of 10 I will lose five tenants

Question

Find the function for the income, I in terms of x, where x is the number of increased

Answer 1

If x is the increase then the new rent is 800+x dollars. The total income would be 500(800+x) if I didn’t lose any tenants. But I lose 5 tenants for every $10 increase. For $1 increase I lose 0.5 tenants. So for an increase of $x I lose x/2 tenants. I started with 500 tenants, assuming 1 tenant per apartment. So the reduction is x/2 leaving 500-x/2 apartments, and I=(500-x/2)(800+x).

To check this out, let x=$50, so I would lose 25 tenants making it 475 apartments and I=475*850=$403750.

Now use the formula: I=(500-25)(800+50)=475*850=$403750. That works!

The formula can be expanded: 400000+100x-x²/2. Substituting x=50 into this gives us $403750. [If the increase were $200 the income would be unchanged at $400000. For an increase below $200 I>$400000; for an increase above $200 I<$400000.]