Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

An electrician is running wire from the electric box on a house to a utility pole 75 feet away. The angle of elevation to the connection on the pole is \(16^{\circ} .\) How much wire does the electrician need?

Step-by-Step Solution

Answer

The electrician needs approximately 77.54 feet long wire.

Step 1: Understanding the Given

Identify and label the sides of the triangle relative to the given angle. Here, the house to pole distance is the adjacent side and the wire is the hypotenuse.

Step 2: Apply Trigonometric Ratio

Choose the correct trigonometric ratio. As we have the adjacent side and we are asked to find the hypotenuse, we use the cosine ratio, which is defined as \(cos(\theta) = \frac{adjacent}{hypotenuse}\). Therefore the length of wire (hypotenuse) can be found using the formula \( wire = \frac{adjacent}{cos( \theta )}\) .

Step 3: Implement and Solve

Substitute the given values into the formula. Thus, \( wire = \frac{75}{cos(16)}\). Calculating the result, approximately, gives us the length of wire required.

Key Concepts

Understanding the Angle of Elevation

The angle of elevation is a fundamental concept in trigonometry. It refers to the angle formed between a horizontal line and the line of sight of an observer looking upwards at an object. In simpler terms, it is the angle you tilt your head back to see the top of something, like a utility pole or a mountain.
The angle of elevation is crucial in right triangle problems, like the one involving the electrician.

• Consider the horizontal distance from the observer to the base of a structure.
• The line of sight forms the hypotenuse of a right triangle.

In our example, the electrician stands on the ground (forming the horizontal or adjacent side), and the wire represents the slanted line (hypotenuse) leading up to the utility pole. The given angle of elevation is used to calculate this length.

Exploring the Cosine Ratio

The cosine ratio is one of the primary trigonometric ratios used to solve right triangle problems. It relates the adjacent side of a right triangle to its hypotenuse. The formula can be expressed as \[cos(\theta) = \frac{adjacent}{hypotenuse}\]This ratio becomes especially useful when you know the length of the adjacent side and need to find the hypotenuse, or vice versa.

• The adjacent side lies next to the angle of elevation.
• The hypotenuse is always the side across from the right angle.

For the electrician running the wire, the adjacent side is the distance from the house to the pole (75 feet), and with the angle of elevation known, the cosine ratio helps determine the length of the wire needed.

Solving Right Triangle Problems

Right triangle problems form some of the most common scenarios in trigonometry, involving calculations around angles and side lengths. They involve triangles with a 90-degree angle, making them easy to handle using trigonometric ratios.

• The Pythagorean theorem often helps when both legs are known and you need to find the hypotenuse.
• The trigonometric ratios—sine, cosine, and tangent—serve to find missing side lengths or angles when only partial information is available.

In the scenario with the electrician, the problem uses the cosine ratio because the length of one side (adjacent to the angle of elevation) and the angle itself are provided. This enables the calculation of how much wire (hypotenuse) is needed by rearranging and solving the cosine formula. Understanding these fundamental methods allows for solving a broad range of real-world triangular problems effectively.