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telnet · Sep 15, 2009, 11:51 AM

The question is: The monthly revenue R from renting x apartments can be modeled by R=2x(900+21x-x^2). If the monthly expense consists of fixed costs totaling $4500, and per unit expense of $150, find:

1. number of rentals that maximize revenue and the profit it produces
2. number of rentals that maximize profit and the profit it produces
3. The monthly apartment rental that maximizes profits.

Idk what I have to do or even how to start?

Thanks :)

ArcSine · Sep 15, 2009, 01:22 PM

The monthly revenue R from renting x apartments can be modeled by R=2x(900+21x-x^2).

First, clarify the revenue function... is that "2" times the expression in parentheses, or is it "2x" times the expression? (It'll make a diff.) Thanks.

telnet · Sep 15, 2009, 04:21 PM

Oh its "2x"

ArcSine · Sep 16, 2009, 05:55 AM

OK, then, multiplying through gives you your Revenue (as a function of x apartment units rented)...

$R(x)\ =\ -2x^3+42x^2+1,800x$

... and your Profit function, which is Revenue less Costs ...

$P(x)\ =\ -2x^3+42x^2+1,650x-4,500$

If you can, graph both of these functions to get an intuitive 'visual' of their behavior as the number of rented units varies.

The answers to your first two questions come from taking the first derivative of each of these functions, and finding those values of x which make the derivatives equal to zero. (Since the functions are cubics, the derivatives will be quadratics, so it amounts to finding the roots of the derivatives, for which you can use factoring, or the Quadratic Formula, or whatever you prefer).

You'll find that they each have two solutions, one negative and one positive. You can immediately disregard the negative solution, since you can't rent a negative number of apartments (even in this economy).

The positive solution for each one corresponds to the number of apt units for which Revenue, and Profit, are maximized, respectively. If you did those graphs, you'll see that those solutions also correspond to the "peaks" of the graphs, which occur in positive-x territory.

As to the third question, I'm not real clear on what's being asked. Can you clarify a bit? (Maybe you were given a "demand" function, showing the number of units rented as a function of the rental rate per unit?)