A given rectangular area is to be fenced off in a field whose length lies along a straight river. If no fencing is needed along the river, find the least amount of fencing will be required when the length of the field is 

The length is twice its breadth $(L=2B)$.

The correct answer is Option (2) → The length is twice its breadth $(L=2B)$.

Let A (positive constant) be the area of the given rectangular field, and $x( > 0)$ be the length of the field, then its breadth = $\frac{A}{x}$.

Let $l$ be the length of the fencing required, then

$l = x + 2.\frac{A}{x}$

Differentiating (i) w.r.t. x, we get

$\frac{dl}{dx}=1-\frac{2A}{x^2}$ and $\frac{d^2l}{dx^2}=\frac{4A}{x^3}$.

Now $\frac{dl}{dx}=0⇒1-\frac{2A}{x^2}=0$

$⇒x^2 = 2A$ but $x > 0 ⇒ x = \sqrt{2A}$.

Also $\left(\frac{d^2l}{dx^2}\right)_{x=\sqrt{2A}}=\frac{4A}{2A.\sqrt{2A}}>0$

$⇒l$ is minimum at $x=\sqrt{2A}$.

Therefore, fencing required is least when length of field = $\sqrt{2A}$

and breadth $=\frac{A}{\sqrt{2A}}=\frac{\sqrt{2A}}{2}=\frac{1}{2}$ of length of field.

Hence, the fencing is least when the length of the field is twice its breadth.