Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Points \(A\) and \(B\) are on opposite sides of Lake Yankee. From a third point, \(C\), the angle between the lines of sight to \(A\) and \(B\) is \(46.3^{\circ} .\) If \(A C\) is 350 meters long and \(B C\) is 286 meters long, find \(A B\).

Step-by-Step Solution

Answer

The distance \( AB \) is approximately 255.42 meters.

Step 1: Identify the Known Values

We are given that \( \angle ACB = 46.3^{\circ} \), \( AC = 350 \) meters, and \( BC = 286 \) meters. We need to find \( AB \), which we can find using the Law of Cosines.

Step 2: Set Up the Law of Cosines

The Law of Cosines states: \( AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(\angle ACB) \). Substitute the known values into this formula.

Step 3: Substitute Known Values into the Equation

Substitute: \( AB^2 = 350^2 + 286^2 - 2 \cdot 350 \cdot 286 \cdot \cos(46.3^{\circ}) \).

Step 4: Calculate \(AB^2\)

Calculate each component: \(350^2 = 122500\), \(286^2 = 81796\), and \(2 \cdot 350 \cdot 286 = 200200\). Evaluate \( \cos(46.3^{\circ}) \) using a calculator.

Step 5: Perform Cosine Calculation and Evaluate the Equation

Use a calculator to find \( \cos(46.3^{\circ}) \approx 0.694658 \). Substitute into the equation: \( AB^2 = 122500 + 81796 - 200200 \times 0.694658 \).

Step 6: Final Calculation for \(AB^2\)

Compute: \( 200200 \times 0.694658 \approx 139027.56 \). Then, \( AB^2 = 204296 - 139027.56 = 65268.44 \).

Step 7: Find \(AB\)

Take the square root of both sides: \( AB = \sqrt{65268.44} \approx 255.42 \) meters.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that deals with the study of triangles, particularly the relationships between the sides and angles of triangles. One key concept in trigonometry is the law of cosines, which is incredibly useful when it comes to solving triangles that are not right-angled. In this exercise involving points A, B, and C, we see a practical application of trigonometry as it helps in calculating the distance across a lake, a problem that requires more than just basic geometry.

The law of cosines is essentially an extension of the Pythagorean theorem applied to any triangle:

• It allows us to find unknown side lengths or angles.
• Helps in solving for a side when two sides and the included angle are known.
• Also aids in angle measurements when all three sides are known.

Understanding trigonometry and its principles is critical because it lays the groundwork for more complex mathematical concepts like calculus and is widely used in fields such as physics, engineering, and even computer graphics.

Angle Calculation

Angle calculation is fundamental in solving geometrical problems. In the given exercise, we start with angle ACB, given as 46.3 degrees. Knowing this angle, along with the lengths of the sides AC and BC, allows us to use the law of cosines to find the length of the side AB.

To perform this calculation, it is important to understand how angles are used in formulas:

• They are often measured in degrees or radians.
• The cosine function is pivotal in translating an angle to a usable scalar for calculations.
• Using a calculator, students can find the cosine of a given angle to plug into trigonometric equations.

In many geometrical problems, accurately calculating angles ensures that the relationships between various components of the figures are correct, making it possible to solve for unknown sides or angles effectively.

Distance Measurement

Distance measurements in a triangle can be efficiently resolved using trigonometric identities like the law of cosines. In the given problem, we need to find the distance AB across Lake Yankee using the known distances AC and BC, alongside the angle between them.

The process of measuring distance requires:

• A clear understanding of involved trigonometric equations.
• Substitution of known values into these equations.
• Performing arithmetic calculations, including the square root to find the actual length.

By plugging the known lengths and angle into the law of cosines, the calculation becomes straightforward, thus determining the exact distance AB. Such techniques are not only useful in classroom exercises but are also essential in real-world applications like construction, navigation, and even in technology for mapping and geographic data systems.