The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 324 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 6 dollar increase in rent. Similarly, one additional unit will be occupied for each 6 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Question

dumbcow · Answer

Revenue = rent*units

rent = 324 + 6x
units = 110 - x

$$R(x) = (324+6x)(110-x) = 35640+336x-6x^{2}$$
to maximize revenue , set derivative equal to zero
$$R'(x) = 336-12x = 0$$
$$x = 336/12 = 28$$
max rent = 324+6*28 = 492

anonymous · Answer

Thank you for your help!

dumbcow · Answer

yw

whpalmer4 · Answer

@mapleryan If you recognize that $R(x)$ is a parabola of the form $ax^2+bx+c$, you can find the x-value of the vertex (point of maximum revenue, in this case) by $x = -b/2a$. Actually the same calculation as with the derivative, but without the need to know the care and feeding of derivatives if you haven't advanced that far yet.