Expert: Scotto - 1/24/2009

Question
Hi,
I have a project in my econ class and I want to wow them with some calculus that you might be able to help me with. We are designing a community and need to make the maximum amount of profit while preserving the maximum amount of acreage. We have three different types of housing that we can build, only 25,000 acres available, and only 390 houses to build.
The house that will make us the most money takes up the most acreage. The second most profitable house takes up medium space, and the third takes up the smallest plot of land.
Is there some possible way to maximize this with optimization or even an equation to help me out a little? Any info or tips on how to go about this project would be gratly appreciated.
Thanks, Jacob.

Answer
In terms of math, the problem is fairly straight forward.

Determine which size house generates the most profit, and build that house all over the place.

Trouble is, we have a limit of 390 houses to build.

Let H = the number of huge houses,
M = the number of medium houses and
S = the number of small houses.

We know that H+M+S <= 390.

Determine which houlse generates the most income.  
If it is the big houses, determine how many fit and build them.
It is probably the small house, though.  Examine the land in
question and see how many of that type house could be built.

If that number tells you to build too many of them, convert many
to the next size up so the number of houses that can be built is
reduced.  I would guess the medium size homes would produce the
next amount of profit per acre.

To make this look like a fine set of houses, it would be good to
put a few large homes in here and there with mediium size homes
distributed all over the place.

In doing this, the maximum profit might not be generated, but the
amount of time to generate that profit would most likely be
increased.