Answer
The length of \[x\] to the nearest whole number is \[298\text{ units}\].
Work Step by Step
In triangle ABC,
Let the side adjacent to \[\text{angle }64{}^\circ \] is \[a\]. Compute the length of \[a\] using the trigonometric function of tangent as follows:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side opposite to angle}\theta }{\text{Adjacent side to angle}\theta }=\frac{AB}{BC} \\
& \text{Tan}{{64}^{\circ }}=\tfrac{300}{a} \\
& 2.050=\tfrac{300}{a}
\end{align}\]
Now, cross multiply to find the value of \[a\]
\[\begin{align}
& a=\tfrac{300}{2.050} \\
& =146.34
\end{align}\]
In triangle ABD,
Let the side adjacent to \[\text{angle }34{}^\circ \] is \[a+x\]. Compute the length of \[x\] using the trigonometric function of tangent as follows:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side opposite to angle}\theta }{\text{Adjacent side to angle}\theta }=\frac{AB}{BD} \\
& \text{Tan}{{34}^{\circ }}=\tfrac{300}{a+x} \\
& 0.674=\frac{300}{146.34+x}
\end{align}\]
On cross multiplying both sides, we get
\[\begin{align}
& 0.674\times \left( 146.34+x \right)=300 \\
& 0.674\times (146.34)+0.674x=300 \\
& 98.633+0.674x=300 \\
& 0.674x=201.367
\end{align}\]
On dividing both sides by \[0.674\], we get
\[\begin{align}
& x=\tfrac{201.367}{0.674} \\
& x=298.76\text{ units}
\end{align}\]
The length of \[x\] to the nearest whole number is \[298\text{ units}\].
