Answer
The length of\[x\]is\[481\text{ units}\].
Work Step by Step
In triangle ABC,
Let the side adjacent to \[\text{angle }43{}^\circ \] is\[a\].
Compute the length of \[x\]using the trigonometric function of Tangent as follows:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}}{\text{Adjacent side to angle}}=\frac{AB}{BC} \\
& \text{Tan}{{43}^{\circ }}=\tfrac{x}{a} \\
& \text{Tan}{{43}^{\circ }}=0.9325 \\
& \tfrac{x}{a}=0.9325
\end{align}\]
On multiplying both sides by\[a\], we get.
\[x=0.9325a\]
In triangle ABD,
Let the side adjacent to \[{{38}^{\circ }}\]is\[a+100\].
Compute the length of the x using the trigonometric function of Tangent as follows:
\[\begin{align}
& \text{Tan}\theta =\frac{\text{Side Opposite to angle}}{\text{Adjacent side to angle}}=\frac{AB}{BD} \\
& \text{Tan}{{38}^{\circ }}=\tfrac{x}{100+a} \\
& 0.7812=\tfrac{0.9325a}{100+a}
\end{align}\]
On cross multiplying both sides, we get
\[\begin{align}
& 0.9325a=0.7812(100+a) \\
& 0.9325a=0.7812\times 100+0.7812a \\
& 0.9325a=78.12+0.7812a \\
& 0.1513a=78.12
\end{align}\]
On dividing both sides by\[0.1513\], we get
\[\begin{align}
& a=\frac{78.12}{0.1513} \\
& =516.325
\end{align}\]
Now, on putting the value of a in the value of x, we get
\[\begin{align}
& x=0.9325a \\
& =0.9325\times 516.325 \\
& =481\text{ units}
\end{align}\]
The length of x to the nearest whole number is\[481\text{ units}\].
