Question 1190183
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Let x be one dimension; then 184/x is the other.<br>
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.<br><hr>
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:<br>
{{{F(x)=3(x)+2(184/x)}}}
{{{F(x)=3x+368/x}}}
{{{dF/dx=3-368/x^2}}}<br>
Set the derivative equal to 0 to solve for x; then finish the problem from there.<br>
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).<br>
<hr>
We can also use an unusual fact about problems like this to be able to set up the problem for solving much more easily.<br><pre>
   +----------------------------------------------------------------------+
   |  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                                         |
   +----------------------------------------------------------------------+</pre>
Using this fact, we can find x far more easily than with the calculus method above.  We only need to solve<br>
{{{3(x)=2(184/x)}}}
{{{3x=368/x}}}
{{{3x^2=368}}}
{{{x^2=368/3}}}
{{{x=sqrt(368/3)=sqrt(1104/9)=(4/3)sqrt(69)}}}<br>
So the total length of fencing in the "x" direction is {{{3((4/3)sqrt(69))=4sqrt(69)}}}<br>
Then we know that the total length of fencing in the other direction is the same, so the total minimum length of fencing is {{{8sqrt(69)}}}<br>
<hr>
You of course should have ended up with the same answer if you finished the solution using calculus.<br>