Calculating Distance Across a River

How to determine the distance across a river using given angles and measurements?

Answer:

To find the distance from point A to point C, we can use the Law of Sines to set up an equation and accurately calculate AC. By plugging in the provided values, we can determine the approximate distance from A to C.

Explaining the Solution:

To find the distance from A to C, we can utilize the Law of Sines, which establishes a relationship between the sides of a triangle and the angles opposite those sides. In this scenario, we are given the side AB and the angles BAC and ABC.

The Law of Sines can be represented by the following formula:
(AC) / (sin(ABC)) = (AB) / (sin(ACB))

By substituting the known values into the equation, we get:
(AC) / (sin(52°)) = 450 / (sin(82°))

Solving for AC, we find that the distance from A to C is approximately 394 feet. Therefore, the correct answer is C) 394 ft.

This calculation showcases the application of trigonometry and the Law of Sines in determining distances across bodies of water, such as rivers. Understanding these mathematical principles is essential for accurate surveying and measurement in various real-world scenarios.