Answer
The buildings are about $257\text{ feet}$ high.
Work Step by Step
Consider that the two buildings of equal height are 800 feet apart and an observer on the street between the buildings measures the angles of elevation to the tops of buildings as $27{}^\circ $ and $41{}^\circ $.
Let h be the height of the buildings.
By observation, we get
$\begin{align}
& b+e=800 \\
& e=800-b
\end{align}$
Using the law of sines, we get
$\frac{b}{\sin 63{}^\circ }=\frac{h}{\sin 27{}^\circ }$ (1)
$\frac{800-b}{\sin 49{}^\circ }=\frac{h}{\sin 41{}^\circ }$ (2)
Now, we will solve equation (1) for b:
$b=\frac{h\sin 63{}^\circ }{\sin 27{}^\circ }$
Substituting the value of b in equation (2):
$\begin{align}
& \frac{800-\left( \frac{h\sin 63{}^\circ }{\sin 27{}^\circ } \right)}{\sin 49{}^\circ }=\frac{h}{\sin 41{}^\circ } \\
& \frac{800\sin 27{}^\circ -h\sin 63{}^\circ }{\sin 27{}^\circ \sin 49{}^\circ }=\frac{h}{\sin 41{}^\circ } \\
& h\left( \sin 27{}^\circ \sin 49{}^\circ \right)=\sin 41{}^\circ \left( 800\sin 27{}^\circ -h\sin 63{}^\circ \right) \\
& h\sin 27{}^\circ \sin 49{}^\circ =800\sin 27{}^\circ \sin 41{}^\circ -h\sin 63{}^\circ \sin 41{}^\circ
\end{align}$
Now, we will solve it further to find h:
$\begin{align}
& h\sin 27{}^\circ \sin 49{}^\circ +h\sin 63{}^\circ \sin 41{}^\circ =800\sin 27{}^\circ \sin 41{}^\circ \\
& h\left( \sin 27{}^\circ \sin 49{}^\circ +\sin 63{}^\circ \sin 41{}^\circ \right)=800\sin 27{}^\circ \sin 41{}^\circ \\
& h=\frac{800\sin 27{}^\circ \sin 41{}^\circ }{\sin 27{}^\circ \sin 49{}^\circ +\sin 63{}^\circ \sin 41{}^\circ } \\
& h\approx 257
\end{align}$
