Question 1146109
<br>
The function for the length of fence is<br>
{{{F(w) = 2w+70/w}}}<br>
Find the value of x that minimizes the area by finding where the derivative is zero.<br>
{{{dF/dw = 2-70/w^2}}}<br>
{{{2-70/w^2 = 0}}}
{{{2 = 70/w^2}}}
{{{2w^2 = 70}}}
{{{w^2 = 35}}}
{{{w = sqrt(35)}}}<br>
The width that minimizes the total amount of fence required is {{{sqrt(35)}}} feet.<br>
That makes the length {{{70/sqrt(35) = (2*35)/sqrt(35) = 2*sqrt(35)}}}<br>
Note that, to minimize the total length of fence required, the length is twice the width.  That is always the case, regardless of what the total area is.<br>
So if you are in a position where you see this kind of problem often -- e.g. you are on a high school math team -- then you can just memorize this fact.<br>
Then to solve this problem without doing the calculus, you just solve<br>
{{{(w)(2w) = 70}}}<br>
leading very quickly to dimensions of {{{sqrt(35)}}} and {{{2*sqrt(35)}}}.