Answer
A square will enclose the maximum area and both the length and breadth of the rectangular region will be \[50\text{ft}\text{.}\]
Work Step by Step
Fencing of \[200\text{ ft}\text{.}\]means that the boundary of the rectangular part is\[200\text{ ft}\text{.}\], that is, the perimeter is\[200\text{ ft}\text{.}\].
Perimeter (P) of a rectangular region can be computed by the below mentioned formula
\[P=2\left( l+b \right)\].
Here l denotes the length and b shows the breadth.
Compute the perimeter as follows:
\[\begin{align}
& P=2\left( l+b \right) \\
& 200=2\left( l+b \right) \\
& 100=l+b
\end{align}\]
So, the sum of length and width is 100 feet. Now, trying with different measurements of length and breadths and calculating the largest area.
If \[l=60\text{ft}\text{.}\]and\[b=40\text{ft}\text{.}\], the area of the rectangle would be:
\[\begin{align}
& A=60\text{ ft}\times 40\text{ ft} \\
& =2400\text{ ft}{{\text{.}}^{\text{2}}}
\end{align}\]
If \[l=70\text{ft}\text{.}\]and\[b=30\text{ft}\text{.}\], the area of the rectangle would be:
\[\begin{align}
& A=70\text{ ft}\times 30\text{ ft} \\
& =2100\text{ ft}{{\text{.}}^{\text{2}}}
\end{align}\]
If \[l=80\text{ft}\text{.}\]and\[b=20\text{ft}\text{.}\], the area of the rectangle would be:
\[\begin{align}
& A=80\text{ ft}\times 20\text{ ft} \\
& =1600\text{ ft}{{\text{.}}^{\text{2}}}
\end{align}\]
If \[l=55\text{ft}\text{.}\]and\[b=45\text{ft}\text{.}\], the area of the rectangle would be:
\[\begin{align}
& A=55\text{ ft}\times 45\text{ ft} \\
& =2475\text{ ft}{{\text{.}}^{\text{2}}}
\end{align}\]
And If \[l=50\text{ft}\text{.}\]and\[b=50\text{ft}\text{.}\], the area of the rectangle would be:
\[\begin{align}
& A=50\text{ ft}\times 50\text{ ft} \\
& =2500\text{ ft}{{\text{.}}^{\text{2}}}
\end{align}\]
So, the area will be highest if the length and the breadth have the same measurements. Therefore, a square will enclose the maximum area.
Hence, a square will enclose the maximum area and both the length and breadth of the rectangular region will be\[50\text{ ft}\text{.}\]
