Question

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, show that the lease length of fencing will be required when length of the field is twice of its breadth.

Answer 1

Let the length and breadth of the rectangular field be l and b respectively.

Since the field is located in such a way that its length lies across a straight river, thus the amount of fencing required can be stated as:-

$f=l+2b$

$\Rightarrow \frac{df}{dl}=1+2\frac{db}{dl}$

The fencing will be minimum or maximum when $\frac{df}{dl}=0$

That is, when $1+2\frac{db}{dl}=0$ -----------(1)

Also, area of the field, A=lb

$\Rightarrow \frac{dA}{dl}=b+l\frac{db}{dl}$

Now, $\frac{dA}{dl}=0whenb+l\frac{db}{dl}=0$

$\Rightarrow \frac{db}{dl}=-\frac{b}{l}$ -------------(2)


Using (2) in (1):-

$$\begin{gathered} 1+2\left(-\frac{b}{l}\right)=0 \\ \Rightarrow l=2b \end{gathered}$$

Hence Proved.