SOLUTION: A 184 m^2 rectangular corn field is to be surrounded by a fence and divided into two equal parts by another fence parallel to two of the sides of the rectangle. What dimensions for
Question 1190183: A 184 m^2 rectangular corn field is to be surrounded by a fence and divided into two equal parts by another fence parallel to two of the sides of the rectangle. What dimensions for the field will require the shortest total length of fencing. What length if fencing will be required? Answer by greenestamps(13200) (Show Source):
Assume there are 3 sections of fence of length x and 2 of length 184/x. Note we could assume the opposite; the numbers in the calculation would be different, but we would end up with the same answer.
We can set up the problem for solving using formal calculus; the objective would be to minimize the total length of 3 sections of fence of length x and 2 of length 184/x:
Set the derivative equal to 0 to solve for x; then finish the problem from there.
I leave the details of that formal solution to you; you should do that, since it will be good exercise. (Obtain an exact solution instead of a decimal approximation).
We can also use an unusual fact about problems like this to be able to set up the problem for solving much more easily.
+----------------------------------------------------------------------+
| For a given area of a rectangular field, the minimum total length |
| of fencing will be when the total lengths of fencing in the |
| two directions are the same |
+----------------------------------------------------------------------+
Using this fact, we can find x far more easily than with the calculus method above. We only need to solve
So the total length of fencing in the "x" direction is
Then we know that the total length of fencing in the other direction is the same, so the total minimum length of fencing is
You of course should have ended up with the same answer if you finished the solution using calculus.
