Rectangle help??

The length of a rectangular garden is double it's width.

The area of the garden is 162 m squared.

If the garden wishes to construct a fence diagonally across the garden,

how much fencing(in meters) will be required?

round your answer to the nearest meter.

Let d = diagonal

Let A = 162 m²

Let L = length

Let w = width

\(\begin{array}{|rcll|} \hline A &=& L\cdot w \quad & | \quad L = 2w \\ A &=& 2w\cdot w \\ A &=& 2w^2 \\ A &=& 2w^2 \quad & | \quad :2 \\ \frac{A}{2} &=& w^2 \\ \mathbf{w^2} & \mathbf{=} & \mathbf{\frac{A}{2}} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline d &=& \sqrt{L^2+w^2} \quad & | \quad L = 2w \\ d &=& \sqrt{(2w)^2+w^2} \\ d &=& \sqrt{4w^2+w^2} \\ d &=& \sqrt{5w^2} \quad & | \quad w^2=\frac{A}{2} \\ d &=& \sqrt{5\cdot \frac{A}{2}} \quad & | \quad A = 162\ m^2 \\ d &=& \sqrt{5\cdot \frac{162}{2}} \\ d &=& \sqrt{5\cdot 81} \quad & | \quad 81 = 9^2\\ d &=& \sqrt{5\cdot 9^2} \\ \mathbf{d} &\mathbf{=}& \mathbf{9\cdot \sqrt{5}\ m} \\ d &=& 20.1246117975\dots\ m \\ d &\approx & 20\ m \\ \hline \end{array} \)

how much fencing(in meters) will be required?  \(20\ m\)

\(\begin{array}{|rcll|} \hline d &=& \sqrt{5w^2} \\ d^2 &=& 5w^2 \\ 5w^2 &=& d^2 \quad & | \quad : 5\\ w^2 &=& \frac{d^2}{5} \\ w &=& \frac{d}{\sqrt{5}} \quad & | \quad 9\cdot \sqrt{5} \\ w &=& \frac{9\cdot \sqrt{5}}{\sqrt{5}} \\ \mathbf{w} &\mathbf{=}& \mathbf{9\ m} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline L &=& 2w \quad & | \quad w = 9 \\ L &=& 2\cdot 9\\ \mathbf{L} &\mathbf{=}& \mathbf{18\ m} \\ \hline \end{array}\)

The length of a rectangular garden is double it's width. The area of the garden is 162 m squared. If the garden wishes to construct a fence diagonally across the garden, how much fencing(in meters) will be required? round your answer to the nearest meter

\(A=l\times w= 2w\times w =162\ m^2\)

 \(2w^2=162m^2\)

\(w=\sqrt \frac{162\ m^2}{2} = 9m\)

\(l=2w=2\times 9m = 18m\)

\(f = \sqrt {w^{2}+l^2}\) = \(\sqrt{(9m)^2+(18m)^2}\)  

\(f=\sqrt{405m^2}\)  = \(20.125\ m\)

\({\color{blue}21\ m \ fencing \ will \ be \ required.}\)

Greeting asinus :- )    !