[Sky's Crying]

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 294 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent. What rent should the manager charge to maximize revenue?

thanks

[flyaway01]

i haven't done optimization or studied calculus in a month now hah.. how much i've forgotten
but i tried and i got $532 per unit
max rev. $40,432
1st equation. 34 less units occupied = 76 units
i may be completely wrong but i felt like trying

[Voltage]

okay you can set this up pretty easily:
to conform with basic calculus,
we'll assign y = revenue, and x = change in price

then y = (294 + x) * (110 - x/7)
simplifying,
y = 32340 + x(110 - 294/7) + x^2/7
y = 32340 - 68x + x^2/7
that's our second order equation that models the conditions
now to optimize:
dy/dx = -68 + 2x/7
0 = -68 + 2x/7
x = 238

Now do the 2nd derivative test for min or max:
d^2(y)/dx^2 = 2/7
it's a max so we found the proper exact solution which is
y(238) = (294 + 238) * (110 - 238/7)
y(238) = 40432
this is the total revenue, how many houses is this? 110 - 238/7 = 76 units

now we got lucky since it just so happens the optimum solution has an integer value number of units but the people that made this question wanted to make things simple