Answer
The length of \[x\] is \[927\text{ units}\].
Work Step by Step
In triangle ABC,
Let the side adjacent to \[48{}^\circ \] is \[a\]. Compute the value of \[a\] using the equation as shown below:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}}{\text{Adjacent side to angle}}=\frac{AB}{BC} \\
& \text{Tan}48{}^\circ =\tfrac{500}{a}\text{ } \\
& 1.11=\tfrac{500}{a}
\end{align}\]
On cross multiplying, we get
\[\begin{align}
& a=\tfrac{500}{1.11} \\
& a=450.45 \\
\end{align}\]
In triangle ABD,
Let the side adjacent to\[{{20}^{\circ }}\] is \[x+a\]. Compute the value of \[x\] using the equation as shown below:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}}{\text{Adjacent side to angle}}=\frac{AB}{BD} \\
& \text{Tan}{{20}^{\circ }}=\tfrac{500}{x+a} \\
& 0.363=\tfrac{500}{x+a}
\end{align}\]
On cross multiplying both sides, we get
\[\begin{align}
& 0.363(x+a)=500 \\
& 0.363(x+450.45)=500 \\
& 0.363x+\left( 0.363\times 450.45 \right)=500 \\
& 0.363x+163.5=500
\end{align}\]
\[\begin{align}
& 0.363x=500-163.5 \\
& 0.363x=336.49
\end{align}\]
On dividing both sides by \[0.363\], we get
\[\begin{align}
& x=\tfrac{336.49}{0.363} \\
& x=926.96\text{ units}
\end{align}\]
Hence, the length of \[x\] to the nearest whole number is \[927\text{ units}\].
