Question

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 288 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

lets imagine that you add 4 $8 increases, then you will make \(288+8\times 4\) dollars per apartment, but only rent \(110-4\) of them for a total income of
\[(288+8\times 4)(110-4)\] dollars

Answer 2

now lets replace \(4\) by \(x\) and say you make \(x\) $8 increases
then you will make \(288+8x\) dollars per apartment and rent \(110-x\) of them for a total of
\[y=(288+8x)(110-x)\]
you want to know where this is largest
multiply out, you get a nice quadratic with negative leading coefficient. it will be largest at the vertex, and the vertex occurs when \(x=-\frac{b}{2a}\)