Let the length and breadth of the rectangular plot be $x$ and $y$ meter, respectively.

Therefore, we have:

$\begin{array}{l}
\text { Perimeter }=2(x+y)=62 \quad \ldots . .(i) \text { and } \\
\text { Area }=x y=228 \\
\Rightarrow y=\frac{228}{x}
\end{array}$

Putting the value of $y$ in (i), we get

$\begin{array}{l}
\Rightarrow 2\left(x+\frac{228}{x}\right)=62 \\
\Rightarrow x+\frac{228}{x}=31 \\
\Rightarrow \frac{x^{2}+228}{x}=31 \\
\Rightarrow x^{2}+228=31 x \\
\Rightarrow x^{2}-31 x+228=0 \\
\Rightarrow x^{2}-(19+12) x+228=0 \\
\Rightarrow x^{2}-19 x-12 x+228=0 \\
\Rightarrow x(x-19)-12(x-19)=0 \\
\Rightarrow(x-19)(x-12)=0 \\
\Rightarrow x=19 \text { or } x=12
\end{array}$

If $x=19 m, y=\frac{228}{19}=12$