Answer
The length of \[x\]is \[257\text{ units}\].
Work Step by Step
In triangle ABC,
Let the side adjacent to angle \[{{40}^{{}^\circ }}\]is\[a\]. Compute the length of \[x\]using the equation as shown below:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}\theta }{\text{Adjacent side to angle}\theta }=\frac{AB}{BC} \\
& \text{Tan}{{40}^{\circ }}=\tfrac{x}{a} \\
& 0.8390=\tfrac{x}{a}
\end{align}\]
On multiplying both sides by\[a\], we get
\[x=0.8390a\]
In triangle ABD,
Now, let the side adjacent to angle\[{{20}^{\circ }}\] is\[a+400\]. Compute the length of \[a\]using the equation as shown below:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}\theta }{\text{Adjacent side to angle}\theta }=\frac{AB}{BD} \\
& \tan {{20}^{\circ }}=\tfrac{x}{a+400} \\
& 0.3639=\tfrac{x}{a+400} \\
& 0.3639=\tfrac{0.8390a}{a+400}
\end{align}\]
On cross multiplying both sides, we get
\[\begin{align}
& 0.3639\times \left( a+400 \right)=0.8390a \\
& \left( 400\times 0.3639 \right)+0.3639a=0.8390a \\
& 145.56+0.3639a=0.8390a \\
& 0.4751a=145.56
\end{align}\]
On dividing both sides by\[0.4751\], we get
\[\begin{align}
& a=\frac{145.56}{0.4751} \\
& =306.37
\end{align}\]
Now, compute the value of \[x\]by putting the value of \[a\]in the value of \[x\] as follows:
\[\begin{align}
& x=0.8390\times a \\
& =0.8390\times 306.37 \\
& =257.050 \\
& =257\text{ units}
\end{align}\]
Hence, the length of x to the nearest whole number is\[257\text{ units}\].
