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

Algebra.Com
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): You can put this solution on YOUR website!


Let x be one dimension; then 184/x is the other.

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.


RELATED QUESTIONS

A rectangular field is to be enclosed by a fence and divided into two parts by another... (answered by ankor@dixie-net.com)
I have 3 questions: 1, a rectangular field is to be enclosed by a fence and divided... (answered by richwmiller)
A farmer wants to make a rectangular field with a total area of . It is surrounded by a... (answered by ankor@dixie-net.com)
A rectangular plot, bounded on one side by a river, is to be enclosed on the other 3... (answered by ankor@dixie-net.com)
A rectangular field is to have an area of 1100 and is to be surrounded by a fence. The... (answered by Fombitz)
A rectangular field is to have an area of 25m^2 and is to be surrounded by a fence. The... (answered by richard1234)
A rectangular field is to be enclosed by a fence and then divded into two small plots by... (answered by ankor@dixie-net.com)
A large open area is to have a section surrounded by a rectangular fence. This rectangle... (answered by ankor@dixie-net.com)
A rectangular field having an area of 2700 m^2 is to be enclosed by a fence, and an... (answered by ankor@dixie-net.com)