Problem 24 - Algebra 2 and Trigonometry

Question

Two points \(A\) and \(B\) are on the shoreline of Lake George. A surveyor is located at a third point \(C\) some distance from both points. The distance from \(A\) to \(C\) is 180.0 meters and the distance from \(B\) to \(C\) is 120.0 meters. The surveyor determines that the measure of \(\angle A C B\) is \(56.3^{\circ} .\) To the nearest tenth of a meter, what is the distance from \(A\) to \(B ?\)

Step-by-Step Solution

Answer

The distance from \(A\) to \(B\) is approximately 151.2 meters.

Step 1: Identify the Known Values

We know the distances from point \(C\) to points \(A\) and \(B\) and the angle \(\angle ACB\). Here are the known values: \(AC = 180.0\) meters, \(BC = 120.0\) meters, and \(\angle ACB = 56.3^{\circ}\). We need to find the distance \(AB\).

Step 2: Apply the Law of Cosines

The Law of Cosines will help us find the unknown side \(AB\) of triangle \(\triangle ABC\). The formula is given by:\[AB^2 = AC^2 + BC^2 - 2 \times AC \times BC \times \cos(\angle ACB)\]Substitute the known values: \(AB^2 = 180.0^2 + 120.0^2 - 2 \times 180.0 \times 120.0 \times \cos(56.3^{\circ})\).

Step 3: Calculate Cosine of the Angle

Calculate \(\cos(56.3^{\circ})\) using a calculator:\[\cos(56.3^{\circ}) \approx 0.5528\]

Step 4: Substitute and Solve for AB

Now substitute \(\cos(56.3^{\circ})\) into the equation from Step 2:\[AB^2 = 180.0^2 + 120.0^2 - 2 \times 180.0 \times 120.0 \times 0.5528\]Calculate each term:\[180^2 = 32400, \quad 120^2 = 14400\]\[2 \times 180 \times 120 \times 0.5528 = 23937.6\]Thus,\[AB^2 = 32400 + 14400 - 23937.6 = 22862.4\]

Step 5: Find the Square Root

Solve for \(AB\) by taking the square root of both sides:\[AB = \sqrt{22862.4} \approx 151.2 \text{ meters}\]

Step 6: Round to the Nearest Tenth

Finally, round \(151.2\) to the nearest tenth. Since it is already at a tenth, no further rounding is needed.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It is particularly powerful when dealing with right-angled triangles but extends to any type of triangle. In trigonometry, one of the most widely used laws for solving triangles that are not right-angled is the Law of Cosines. This law provides a way to find unknown sides or angles in a triangle when certain parts are known. In our problem, we applied this concept to find the unknown distance between points in a real-world setting.

• The Law of Cosines is essential when dealing with oblique triangles, where not all angles are 90 degrees.
• Trigonometry involves the study of various functions such as sine, cosine, and tangent, which relate the angles to side lengths.
• This area of mathematics is crucial for various applications, from navigation and architecture to physics and engineering.

Understanding the basics of trigonometry prepares you to tackle complex triangle problems like the one in this exercise.

Triangle Geometry

Triangle geometry deals with the properties and types of triangles. The key focus is on understanding the relationships between different sides and angles. In any given triangle, the sum of the interior angles is always 180 degrees. Knowing any two angles allows the calculation of the third.

• Triangles can be classified based on their angles, such as acute, obtuse, or right-angled.
• They can also be classified by their sides: equilateral (all sides equal), isosceles (two sides equal), or scalene (no sides equal).
• The Law of Cosines is particularly useful in scalene triangles, where all sides and angles are different.

This fundamental understanding of triangle properties helps in calculating unknown sides or angles, such as determining the distance from point A to point B in our problem.

Distance Calculation

Distance calculation plays a crucial role in many real-world situations. It often involves using known measurements and applying mathematical laws to find unknown lengths. The Law of Cosines is a powerful tool in such scenarios, especially when direct measurement isn't possible.

• The formula for the Law of Cosines is \[c^2 = a^2 + b^2 - 2ab \times \cos(C)\]. This relates the lengths of sides in a triangle to the cosine of one of its angles.
• In our exercise, this formula helped calculate the unknown distance between two points, necessary for surveyors and engineers.
• Understanding how to manipulate this formula is key in accurately determining distances from various known points.

By appreciating the principles behind distance calculations, you can solve practical problems effectively, as demonstrated in how we derived the distance from point A to point B.