What rent should the manager charge to maximize revenue? (Optimization)

Question

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 348 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?

Answer 1

To find the optimum rent (that is maximum revenue), we have to set up a quadratic equation follows:

Revenue, y = rent x units rented. Based on the information given, the equation will be:

y = (348 + 6x) (120 - x)

where x is the integer change from 120 in units rented. It will be negative for increased rent and positive for lower rent.

So the quadratic is:

y = -6x^2 + 372x + 41760 (a=-6, b=372, c=41760)

The maxima of this parabola is given by: -b/2a = -372/2(-6) = 31

This means, the optimum units rented = 120 - 31 =89

At that level, the rent he needs to charge = 348 + 6 x 31 = $534

He will rent out only 89 units, but his revenue will be maximum