https://file.aiquickdraw.com/mathbot/file/1729492057077yi7knt8v.jpg

The problem asks to find the distance between points \( A \) and \( B \) across the lake. We are given:

- Angle \( \angle CAB = 50.8^\circ \)
- Side \( AC = 358 \, \text{m} \)
- Side \( BC = 471 \, \text{m} \)

This forms a triangle, and we need to find side \( AB \).

We can use the **Law of Cosines** to solve for \( AB \). The formula for the Law of Cosines is:

\[
AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(\angle CAB)
\]

Substituting the given values:

\[
AB^2 = 358^2 + 471^2 - 2 \cdot 358 \cdot 471 \cdot \cos(50.8^\circ)
\]

Let's calculate this step-by-step.

1. \( 358^2 = 128,164 \)
2. \( 471^2 = 221,241 \)
3. \( \cos(50.8^\circ) \approx 0.6336 \)
4. \( 2 \cdot 358 \cdot 471 \cdot 0.6336 \approx 213,878.77 \)

Now, applying these to the Law of Cosines:

\[
AB^2 = 128,164 + 221,241 - 213,878.77
\]
\[
AB^2 = 135,526.23
\]
\[
AB = \sqrt{135,526.23} \approx 368.12 \, \text{m}
\]

Thus, the distance across the lake from \( A \) to \( B \) is approximately \( 368.12 \, \text{m} \).

Would you like any more details or have any questions?

Here are five related questions:
1. What is the Law of Cosines, and when is it applicable?
2. How do you calculate the cosine of an angle in degrees?
3. Can the Law of Cosines be used to find angles if all sides are known?
4. What is the relationship between the Law of Sines and the Law of Cosines?
5. How does the accuracy of angle measurements affect the result?

**Tip:** Always ensure your calculator is set to degrees (not radians) when working with angles measured in degrees.

Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such that ∠CAB = 50.8°. Find the distance across the lake from A to B. AC = 358 m, BC = 471 m.

Learn how to find the distance between points A and B across a lake using the Law of Cosines. This problem involves calculating side AB of a triangle with angle ∠CAB = 50.8°, AC = 358 m, and BC = 471 m. Suitable for high school trigonometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation