Problem 34 - Algebra and Trigonometry

Question

Distance Across a Lake 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 \(\angle C A B=48.6^{\circ} .\) He also measures \(C A\) as 312 \(\mathrm{ft}\) and \(C B\) as 527 ft. Find the distance between \(A\) and \(B\)

Step-by-Step Solution

Answer

The distance between \( A \) and \( B \) is approximately 401.28 feet.

Step 1: Identify the triangle

In this problem, we have a triangle \( \triangle ABC \) where points \( A \) and \( B \) are separated by the lake, and \( C \) is on land. We know two sides (\( CA = 312 \) ft and \( CB = 527 \) ft) and the included angle (\( \angle CAB = 48.6^{\circ} \)).

Step 2: Apply the Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For \( \triangle ABC \), we apply:\[AB^2 = CA^2 + CB^2 - 2 \times CA \times CB \times \cos(\angle CAB)\]

Step 3: Calculate \( AB^2 \)

Substitute the given values into the Law of Cosines formula:\[AB^2 = 312^2 + 527^2 - 2 \times 312 \times 527 \times \cos(48.6^{\circ})\]\[AB^2 = 97344 + 277729 - 2 \times 312 \times 527 \times \cos(48.6^{\circ})\]Calculate these components step by step.

Step 4: Compute the cosine term

Calculate the cosine of the angle:\[\cos(48.6^{\circ}) \approx 0.6561\]So, the term becomes:\[2 \times 312 \times 527 \times 0.6561 = 214048.89\]

Step 5: Final calculation of \( AB^2 \)

Calculate \( AB^2 \) using the computed term:\[AB^2 = 97344 + 277729 - 214048.89 = 161024.11\]

Step 6: Solve for \( AB \)

Take the square root of \( AB^2 \) to find \( AB \):\[AB = \sqrt{161024.11} \approx 401.28 \text{ ft}\]

Step 7: Conclusion

The distance between points \( A \) and \( B \), across the lake, is approximately 401.28 feet.

Key Concepts

Triangle Properties

Understanding the properties of triangles is crucial in solving many geometric problems. Triangles are three-sided polygons with three angles. The sum of the angles in any triangle is always equal to 180 degrees.
In the given problem, we have triangle \( \triangle ABC \) with known sides \( CA \) and \( CB \), and their included angle \( \angle CAB \). This forms a Side-Angle-Side (SAS) configuration, which is ideal for applying the Law of Cosines.
Moreover, properties like congruency and similarity help establish other triangle relationships, although they aren't directly used in our current problem. Recognizing SAS helps us understand why the Law of Cosines is our tool here.

Trigonometry

Trigonometry is a key mathematical concept that deals with the relationships between the angles and sides of triangles. It forms the backbone of methods like the Law of Sines and the Law of Cosines.
For \( \triangle ABC \), the Law of Cosines is crucial. It serves as an extension of the Pythagorean theorem to non-right triangles, allowing us to integrate an angle into our calculations. The formula: \[AB^2 = CA^2 + CB^2 - 2 \times CA \times CB \times \cos(\angle CAB)\] helps us find the unknown side when two sides and their included angle are known.
This technique links geometric and algebraic concepts and demonstrates the power of trigonometric functions like cosine in measuring distances not easily accessible.

Surveying Techniques

Surveying techniques involve various methods for determining distances and angles between points, often for construction or land measurement purposes. The scenario of measuring the distance across a lake showcases a classic surveying application.
In practice, a surveyor might use tools like a theodolite to accurately measure angles, and chaining or electronic distance measurement (EDM) devices to measure distances, especially in situations where direct measurement is impossible, such as across a body of water.

• By choosing a point \( C \) on accessible land, the task becomes feasible. Knowing the distances \( CA \) and \( CB \), along with the angle \( \angle CAB \), facilitates the use of trigonometric principles.
• The Law of Cosines becomes the practical solution here, converting indirect measurements into a direct distance.

Understanding these techniques underscores the real-world applications of trigonometry in fields like surveying, ensuring precision in landscape mapping.