Answer
The platform is about $3671.8\text{ yards}$ from one end of the beach and $\text{3576}\text{.4 yards}$ from the other end.
Work Step by Step
Using the figure, we will find C:
$\begin{align}
& C=180{}^\circ -A-B \\
& =180{}^\circ -85{}^\circ -76{}^\circ \\
& =19{}^\circ
\end{align}$
Now, the ratio $\frac{c}{\sin C}$, or $\frac{1200}{\sin 19{}^\circ }$, is known.
Using the law of sines we will find a and b:
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{a}{\sin 85{}^\circ }=\frac{1200}{\sin 19{}^\circ } \\
& a=\frac{1200\sin 85{}^\circ }{\sin 19{}^\circ } \\
& a\approx 3671.8
\end{align}$
Now, we will evaluate b using the law of sines as below:
$\begin{align}
& \frac{b}{\sin B}=\frac{c}{\sin C} \\
& \frac{b}{\sin 76{}^\circ }=\frac{1200}{\sin 19{}^\circ } \\
& b=\frac{1200\sin 76{}^\circ }{\sin 19{}^\circ } \\
& b\approx 3576.4
\end{align}$
The platform is about $3671.8\text{ yards}$ from one end of the beach and $\text{3576}\text{.4 yards}$ from the other end.
