Question

a triangular delta has been formed at the mouth of a large river by sediment deposits. the distance from the coast to the starting point of the delta is 80 km, and the angles formed by the coastline and the two sides of the triangular delta are 58° and 47°. determine the length of the delta along the coastline. coast c 58° 47° 80 km θ

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is $180^{\circ}$. Let $\theta$ be the third - angle. So, $\theta=180^{\circ}-(58^{\circ}+47^{\circ}) = 75^{\circ}$.

Step2: Use the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Let the side opposite the $58^{\circ}$ angle be $a$, the side opposite the $47^{\circ}$ angle be $b$, and the side opposite the $75^{\circ}$ angle be $c = 80$ km.
We want to find the side opposite the $58^{\circ}$ angle. Using the Law of Sines $\frac{a}{\sin58^{\circ}}=\frac{80}{\sin75^{\circ}}$.
So, $a=\frac{80\times\sin58^{\circ}}{\sin75^{\circ}}$.
Since $\sin58^{\circ}\approx0.848$ and $\sin75^{\circ}\approx0.966$, then $a=\frac{80\times0.848}{0.966}\approx70.3$ km.
We also want to find the side opposite the $47^{\circ}$ angle. Using the Law of Sines $\frac{b}{\sin47^{\circ}}=\frac{80}{\sin75^{\circ}}$.
So, $b = \frac{80\times\sin47^{\circ}}{\sin75^{\circ}}$.
Since $\sin47^{\circ}\approx0.731$ and $\sin75^{\circ}\approx0.966$, then $b=\frac{80\times0.731}{0.966}\approx60.5$ km.

Answer:

The lengths of the two sides of the triangular delta are approximately $70.3$ km and $60.5$ km.