Problem 44 - Contemporary Precalculus

Question

A triangular piece of land has two sides that are 80 feet and 64 feet long, respectively. The 80 -foot side makes an angle of \(28^{\circ}\) with the third side. An advertising firm wants to know whether a 30 -foot long sign can be placed along the third side. What would you tell them?

Step-by-Step Solution

Answer

Answer: Yes, the sign can be placed along the third side of the triangular piece of land because the third side is approximately 38.12 ft long, which is longer than the 30-foot long sign.

Step 1: Identify the given information

We have a triangle with sides A (80 ft), B (64 ft), and an angle C (28 degrees) between them. We need to find the length of side C.

Step 2: Apply the Law of Cosines

The Law of Cosines states that \(c^2 = a^2 + b^2 - 2ab \cdot cos(C)\), where a, b, and c are the lengths of the sides of the triangle and C is the angle between sides a and b. In this case, a=80 ft, b=64 ft, and C=28 degrees.

Step 3: Calculate the length of side C

Substitute the given values into the Law of Cosines formula: \(c^2 = 80^2 + 64^2 - 2 \cdot 80 \cdot 64 \cdot cos(28^{\circ})\) First, find the cosine of 28 degrees, which is approximately 0.8829. \(c^2 = 80^2 + 64^2 - 2 \cdot 80 \cdot 64 \cdot 0.8829\) Calculate the value of each term: \(6400 + 4096 - 9042.112\) Now, add and subtract the terms: \(c^2 = 1453.888\) Lastly, take the square root of both sides to find the length of side c: \(c ≈ \sqrt{1453.888} ≈ 38.12\) ft

Step 4: Compare the length of side C with the length of the sign

The length of side C is approximately 38.12 ft, which is longer than the 30-foot long sign. Therefore, the sign can be placed along the third side of the triangular piece of land.

Key Concepts

Triangles

Triangles are fascinating shapes and are fundamental in geometry. A triangle is a polygon with three edges and three vertices.
They can be classified based on their sides and angles:

• Equilateral triangle: All sides and angles are equal.
• Isosceles triangle: Two sides are equal, and two angles are the same.
• Scalene triangle: All sides and angles are different.
• Right triangle: One angle is exactly 90 degrees.

Understanding triangles helps solve many geometric problems, like the one in this exercise, where knowing how a side can determine the placement of an object along its length is crucial.

Geometry Problems

Solving geometry problems involves understanding different shapes and their properties. In this context, you often need to calculate unknown values using given data, like side lengths or angles.
In our exercise, we were given some sides and an angle of a triangle and were asked if a given length can fit along an unknown side.
Using these clues, traditional mathematical methods such as the Law of Cosines can find missing measures and solve the problem. Diving deeper into geometry problems requires practice in recognizing known patterns and applying relevant formulas.

Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles of triangles.
It plays a crucial role in solving problems related to the measurement of angles and the computation of lengths.
Key concepts in trigonometry include:

• Sine, Cosine, and Tangent: Ratios that relate angles to side lengths.
• The Unit Circle: A circle with a radius of 1, used to define trigonometric functions.
• Pythagorean Identity: Relates the squares of sine and cosine functions.

In this exercise, the Law of Cosines, which is rooted in trigonometric principles, helped identify the unknown side length.

Angle Measurement

Understanding angle measurement is an integral part of geometry and trigonometry. Angles are measured in degrees or radians, and they describe the amount of rotation required to superpose one of two intersecting lines onto the other.
Measuring angles accurately is essential for problems like this, where knowing the angle helps determine the side lengths of a triangle. Some common types of angles include:

• Acute angle: Less than 90 degrees.
• Right angle: Exactly 90 degrees.
• Obtuse angle: More than 90 but less than 180 degrees.

For the Law of Cosines to be applied correctly in our exercise, the angle between the known sides must be precisely measured in degrees.

Side Length Calculation

Calculating side lengths in triangles often requires using known formulas and the information given. In our exercise, we used the Law of Cosines formula \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\) to calculate the unknown side.
By substituting the given values into the formula, we used arithmetic operations to find the desired length, proving that this method is reliable for such computations.
Steps generally include:

• Substitute known lengths and angle into the Law of Cosines.
• Perform arithmetic operations to solve for the unknown.
• Take the square root of the result to find the side length.

Successfully calculating side lengths allows us to confirm possible real-world applications, such as placing a sign along a specific triangle side.