An apartment complex contains 250 apartments to rent. If they rent x apartments then their monthly profit is specified by, in dollars,, P ( x ) = -8x^{2} + 3200x - 80, 000

How several apartments must they rent in order to maximize their profit?

**Solution : **All that we're actually being asked to do is to maximize the profit subject to the constraint that x must be in the range 0 ≤ x ≤ 250 .

Firstly, we'll required the derivative & the critical point(s) which fall in the range 0 ≤ x ≤ 250

P′ (x) =-16x + 3200 ⇒ 3200 -16 x = 0 ⇒ x = 3200/ 16 = 200

As the profit function is continuous and we have an interval with finite bounds we can determine the maximum value through simply plugging in the only critical point which we have (that nicely enough in the range of acceptable answers) and the ending points of the range.

P (0) = -80, 000 P ( 200) =240, 000 P ( 250) = 220, 000

therefoer, it looks like they will produce the most profit if only they rent out 200 of the apartments rather than all 250 of them.

There are couples of very real applications to calculus which are in the business world and at some level i.e. the point of this section. Note as well that to actually learn these applications and all of their intricacies you'll have to take a business course or two or three. In this section we're only going to scratch the surface & get a feel for some of the real applications of calculus from the business world and some main "buzz" words in the applications.