Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Advertising sign height Added in 1990 and removed in 1997 , the highest advertising sign in the world was a large letter I situated at the top of the 73-story First Interstate World Center building in Los Angeles. At a distance of 200 feet from a point directly below the sign, the angle between the ground and the top of the sign was \(78.87^{\circ}\). Approximate the height of the top of the sign.

Step-by-Step Solution

Answer

The approximate height of the top of the sign is 965.2 feet.

Step 1: Understand the Problem

We need to find the height above the ground of the advertising sign. We are given a right triangle where one leg is the horizontal distance (200 feet) and the angle of elevation is \( 78.87^\circ \). The height of the sign forms the opposite side of this triangle.

Step 2: Recall Trigonometric Functions

For a right triangle, the tangent of an angle \( \theta \) is equal to the ratio of the opposite side to the adjacent side. Here, the opposite side is the height of the sign, and the adjacent side is the horizontal distance (200 feet). The formula is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Step 3: Set Up the Equation

Using the tangent formula: \( \tan(78.87^\circ) = \frac{h}{200} \), where \( h \) is the height of the top of the sign. We will solve for \( h \).

Step 4: Solve for Height

Calculate \( \tan(78.87^\circ) \) using a calculator to find the height \( h \). Then solve the equation: \[ h = 200 \times \tan(78.87^\circ) \]

Step 5: Calculate Tan Value

Using a calculator, find \( \tan(78.87^\circ) \). Make sure your calculator is set to degree mode. You should get approximately 4.826.

Step 6: Compute the Height

Substitute the tangent value into the equation: \[ h = 200 \times 4.826 \]Calculate the product to find \( h \), the approximate height of the sign.

Step 7: Final Calculation

Perform the multiplication:\[ h = 200 \times 4.826 = 965.2 \]Thus, the approximate height of the top of the sign is 965.2 feet.

Key Concepts

Angle of Elevation

The concept of the angle of elevation is a fundamental idea in trigonometry, especially useful in various practical scenarios. An angle of elevation is formed when you look upward from a horizontal line to a point above the line, creating an angle. Imagine standing 200 feet away from a towering sign and tilting your head up to view the top; this tilt creates the angle of elevation. In problems involving height, like finding the height of a sign, the angle of elevation serves as an essential clue. By knowing this angle, you can use trigonometric functions to calculate unseen heights.

Applications of the angle of elevation include engineering, architecture, and even simple everyday tasks. Whenever you need to determine the height of an object without directly measuring it, understanding and using the angle of elevation can make the task feasible, ensuring you have all the right tools to solve real-world puzzles.

Tangent Function

The tangent function is a primary trigonometric function, often used when dealing with right triangles, particularly in vertical height problems like finding the height of a sign. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. In the context of our exercise, you are given the angle of elevation and need to find the height, which forms the opposite side of your right triangle.

When you know the angle and one side, such as the ground distance (adjacent side), you can calculate the length of the opposite side using the tangent function as follows:

• The formula for tangent is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• By rearranging this formula, you can solve for the unknown opposite side: \( \text{opposite} = \tan(\theta) \times \text{adjacent} \).

With this simple yet powerful formula, you can now tackle similar problems, knowing that the tangent function provides a direct path to solving height calculations when an angle of elevation is involved.

Right Triangle

A right triangle is a type of triangle where one of its angles is exactly 90 degrees. In many trigonometric problems, like the one we've studied, right triangles are indispensable. They provide a structured way to employ trigonometric functions to find unknown sides or angles.

In a right triangle, remember:

• One angle is always 90 degrees.
• The two sides forming this angle are referred to as the adjacent and opposite sides when considering a given angle other than the right angle.
• These sides and the hypotenuse (the side opposite the right angle) relate through trigonometric functions like sine, cosine, and tangent.

Understanding how to identify and work with right triangles simplifies many mathematical problems, particularly when dealing with angles and distances. Right triangles offer a foundation for calculations in trigonometry, ensuring you can make precise mathematical deductions from a given set of measurements.