To approximate the distance from point $A$ to point $C$ across the river, we can apply the Law of Sines. The Law of Sines relates the angles and sides of a triangle.

Given:

• $AB = 300 \, \text{ft}$
• $\angle BAC = 82^\circ$
• $\angle ABC = 52^\circ$

First, we can calculate the third angle $\angle ACB$: $\angle ACB = 180^\circ - \angle BAC - \angle ABC = 180^\circ - 82^\circ - 52^\circ = 46^\circ$

Now, by using the Law of Sines: $\frac{AC}{\sin(\angle ABC)} = \frac{AB}{\sin(\angle ACB)}$

Substitute the known values: $\frac{AC}{\sin(52^\circ)} = \frac{300 \, \text{ft}}{\sin(46^\circ)}$

Now, solve for $AC$: $AC = \frac{300 \cdot \sin(52^\circ)}{\sin(46^\circ)}$

Let's compute the value.The approximate distance from point $A$ to point $C$ is 329 feet.

Would you like more details on the calculation process or have any further questions?

Here are 5 related questions to expand on this topic:

1. How would the solution change if the distance between $A$ and $B$ was different?
2. How can the Law of Cosines be used to solve similar problems?
3. What happens if all angles of the triangle are given, but no sides?
4. How can the Law of Sines be used to find an angle, given two sides and an angle?
5. How would you calculate the area of the triangle formed by points $A$, $B$, and $C$?

Tip: Always check if a problem involves solving a triangle using either the Law of Sines or Law of Cosines, based on the information provided.