Problem 36 - Precalculus Mathematics for Calculus

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 ft and \(C B\) as 527 ft. Find the distance between \(A\) and \(B\).

Step-by-Step Solution

Answer

The distance between points A and B is approximately 396 ft.

Step 1: Use of Law of Cosines

To find the distance between points A and B across the lake, we use the Law of Cosines. For triangle \( \triangle CAB \), the Law of Cosines states: \( AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(\angle CAB) \).

Step 2: Substitute Known Values

From the problem, we know that \( \angle CAB = 48.6^\circ \), \( AC = 312 \) ft, and \( BC = 527 \) ft. Substitute these values into the equation: \( AB^2 = 312^2 + 527^2 - 2 \cdot 312 \cdot 527 \cdot \cos(48.6^\circ) \).

Step 3: Calculate \( AB^2 \)

First, calculate \( 312^2 = 97344 \) and \( 527^2 = 277729 \). Next, calculate \( \cos(48.6^\circ) \) using a calculator, which is approximately 0.6626. Then, compute \( 2 \cdot 312 \cdot 527 \cdot 0.6626 = 218233.1664 \).

Step 4: Solve for \( AB \)

Now substitute these values back into the equation: \( AB^2 = 97344 + 277729 - 218233.1664 \). This simplifies to \( AB^2 = 156839.8336 \). Finally, take the square root to find \( AB = \sqrt{156839.8336} \approx 396 \) ft.

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool in trigonometry for solving triangles that are not right-angled. It helps us find unknown side lengths or angles in a triangle when we have enough information. Think of it as an extended version of the Pythagorean theorem but for any type of triangle, not just right triangles.

The Law of Cosines states that for any triangle with sides labeled as \( a \), \( b \), and \( c \), and with \( C \) being the angle opposite side \( c \), the relationship is given by:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

This formula allows you to find the length of one side if you know the lengths of the other two sides and the measure of the included angle.

This rule is particularly useful when dealing with real-life problems, such as the one in our exercise where a surveyor needs to find the distance across a lake using known measurements and an angle.

Triangle

Triangles are everywhere in geometry, and understanding them is crucial to solving many geometry problems. A triangle is a simple polygon with three sides and three angles. They come in different types, mainly categorized by their angles or their sides.

For instance:

• Equilateral Triangle: All sides and all angles are equal.
• Isosceles Triangle: Two sides are of equal length, and their opposite angles are equal.
• Scalene Triangle: All sides and angles are different from one another.
• Right Triangle: One of the angles is exactly 90 degrees.

In our specific example, we are dealing with a scalene triangle, where all side lengths and angles are different. With given sides and an angle, as seen in the problem, we use trigonometry to find unknown measurements. Understanding the fundamental aspects of triangles allows us to apply rules and formulas accurately.

Angle Measurement

Angles are a key component of many geometric problems, including those involving triangles. An angle is formed by two rays meeting at a vertex and is measured in degrees. Knowing how to measure angles helps to apply trigonometric rules effectively.

In our exercise, the angle \( \angle CAB \) was given as 48.6 degrees. Measuring angles accurately is vital because even a small error can lead to incorrect results in calculations.

There are several ways to measure angles:

• Degrees: The most common unit, where a full circle is 360 degrees.
• Radians: Used mainly in higher mathematics, where a full circle is \( 2\pi \) radians.

In trigonometry, we often rely on a calculator to find the cosine of an angle, as done in this exercise. Accurate angle measurement and understanding are fundamental in ensuring that we solve problems effectively and precisely.