# Business Applications - Business Applications In the final...

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Business Applications In the final section of this chapter let’s take a look at some applications of derivatives in the business world. For the most part these are really applications that we’ve already looked at, but they are now going to be approached with an eye towards the business world. Let’s start things out with a couple of optimization problems. We’ve already looked at more than a few of these in previous sections so there really isn’t anything all that new here except for the fact that they are coming out of the business world. Example 1 An apartment complex has 250 apartments to rent. If they rent x apartments then their monthly profit, in dollars, is given by, How many apartments should they rent in order to maximize their profit? Solution All that we’re really being asked to do here is to maximize the profit subject to the constraint that x must be in the range . First, we’ll need the derivative and the critical point(s) that fall in the range . Since the profit function is continuous and we have an interval with finite bounds we can find the maximum value by simply plugging in the only critical point that we have (which nicely enough in the range of acceptable answers) and the end points of the range. So, it looks like they will generate the most profit if they only rent out 200 of the apartments instead of all 250 of them.

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Note that with these problems you shouldn’t just assume that renting all the apartments will generate the most profit. Do not forget that there are all sorts of maintenance costs and that the more tenants renting apartments the more the maintenance costs will be. With this analysis we can see that, for this complex at least, something probably needs to be done to get the maximum profit more towards full capacity. This kind of analysis can help them determine just what they need to do to move towards goal that whether it be raising rent or find a way to reduce maintenance costs. Note as well that because most apartment complexes have at least a few unit empty after a tenant moves out and the like that it’s possible that they would actually like the maximum profit to fall slightly under full capacity to take this into account. Again,
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