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Section 4.14 : Business Applications

2. A management company is going to build a new apartment complex. They know that if the complex contains \(x\) apartments the maintenance costs for the building, landscaping etc. will be,

\[C\left( x \right) = 4000 + 14x - 0.04{x^2}\]

The land they have purchased can hold a complex of at most 500 apartments. How many apartments should the complex have in order to minimize the maintenance costs?

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Start Solution

Because these are essentially the same type of problems that we did in the Absolute Extrema section we will not be doing a lot of explanation to the steps here. If you need some practice on absolute extrema problems you should check out some of the examples and/or practice problems there.

All we really need to do here is determine the absolute minimum of the maintenance function and the value of \(x\) that will give the absolute minimum.

Here is the derivative of the maintenance function and the critical point(s) since we’ll need those for this problem.

\[C'\left( x \right) = 14 - 0.08x = \hspace{0.5in}\, \Rightarrow \hspace{0.5in}\,\,x = 175\] Show Step 2

From the problem statement we can see that we only want critical points that are in the interval \(\left[ {0,500} \right]\). As we can see both of the critical points from the above step are in this interval and so we’ll need both of them.

Show Step 3

The next step is to evaluate the maintenance function at the critical points from the second step and at the end points of the given interval. Here are those function evaluations.

\[C\left( 0 \right) = 4000\hspace{0.5in}C\left( {175} \right) = 5225\hspace{0.75in}C\left( {500} \right) = 1000\] Show Step 4

From these evaluations we can see that the complex should have 500 apartments to minimize the maintenance costs.