Answer
The value of $h=481.6$.
Work Step by Step
In the provided graph
$\begin{align}
& \angle ABC=180{}^\circ -67{}^\circ \\
& =113{}^\circ \\
& \angle ACB=180{}^\circ -43{}^\circ -113{}^\circ \\
& =24{}^\circ
\end{align}$
Using the law of sines we will find $\overline{BC}$
$\begin{align}
& \frac{\overline{BC}}{\sin 43{}^\circ }=\frac{312}{\sin 24{}^\circ } \\
& \overline{BC}=\frac{312\sin 43{}^\circ }{\sin 24{}^\circ } \\
& \overline{BC}\approx 523.1
\end{align}$
Now, we will use the law of sines to find h.
$\begin{align}
& \frac{h}{\sin 67{}^\circ }=\frac{523.1}{\sin 90{}^\circ } \\
& h=\frac{523.1\sin 67{}^\circ }{\sin 90{}^\circ } \\
& h\approx 481.6
\end{align}$
Hence, the value of $h=481.6$.
