Problem 26 - Precalculus: Functions and Graphs

Question

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 top of the sign is approximately 974.6 feet above the ground.

Step 1: Understanding the Problem

We are given a point that is directly below the sign. From a distance of 200 feet horizontally from this point, we observe an angle of elevation of \(78.87^{\circ}\) to the top of the sign. Our task is to determine the height of the sign above the ground.

Step 2: Drawing a Right Triangle

Visualize a right triangle where one vertex is at the base of the building (point directly below the sign), another vertex is at the observation point 200 feet away, and the third vertex is at the top of the sign. The angle of \(78.87^{\circ}\) is between the ground and the line of sight to the top of the sign.

Step 3: Using Trigonometry to Solve

In the right triangle, the horizontal distance (adjacent side) is 200 feet, and the angle of elevation is \(78.87^{\circ}\). The height of the sign (opposite side) can be found using the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Step 4: Calculating the Height

Substitute the known values into the tangent equation: \( \tan(78.87^{\circ}) = \frac{x}{200} \), where \(x\) is the height of the sign. Solve for \(x\) by multiplying both sides by 200: \( x = 200 \times \tan(78.87^{\circ}) \).

Step 5: Performing the Calculation

Using a calculator, find \( \tan(78.87^{\circ}) \approx 4.873 \). Then calculate: \( x = 200 \times 4.873 \approx 974.6 \) feet.

Key Concepts

Angle of Elevation

The angle of elevation is an important concept in trigonometry. Imagine standing at a point on the ground and looking up at a high object, like the top of a building. The angle of elevation is the angle between the horizontal ground line and your line of sight looking upwards. In simpler terms, it's how high from the ground your eyes need to look to see the top of something.
For example, in the problem given, the angle of elevation is specified as \(78.87^{\circ}\). With this angle, one can look up from a point on the ground to see the top of a sign situated on a building. This angle helps determine the other measurements needed to solve the problem using trigonometry.
Knowing how to find the angle of elevation allows you to calculate heights and distances without having to measure them directly. It's a crucial step in problems involving heights.

Tangent Function

The tangent function is one of the primary trigonometric functions and is key to solving many real-world problems. Tangent, often abbreviated as \( \tan \), relates the angles and sides of a right triangle. Specifically, in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.

• For any angle \( \theta \) in a right triangle:
• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In our example, the problem gives an angle of elevation and a known adjacent side of the triangle, which is the distance from the point directly below the sign to the observation point, 200 feet away.
The formula allows us to find the opposite side, or the height of the building sign in this context. The tangent function is essential because it ties together the angle and the sides, making it easy to calculate heights or distances using basic trigonometric principles.

Right Triangle

Understanding the right triangle is crucial in many trigonometric calculations. In trigonometry, a right triangle is a triangle with one 90-degree angle. It is defined by three sides: the opposite, the adjacent, and the hypotenuse.

• The hypotenuse is the longest side opposite the right angle.
• The opposite side is the one facing a given angle of interest, in this case, the line of sight to the top of the sign.
• The adjacent side is the bottom side that lies next to the angle of interest, commonly the ground distance in these scenarios.

In the problem at hand, the right triangle helps in visualizing the relationship between these sides and the angle of elevation. This mental visualization is composed of the vertical line representing the building height and the horizontal line of 200 feet, which together with the angle of elevation forms a right triangle. Knowing these aspects allows for the effective use of trigonometry to solve the problem.

Height Calculation

Height calculation using trigonometry hinges on understanding angles and relationships between triangle sides. With the angle of elevation and a known horizontal distance, it's possible to calculate the height of an object using the tangent function.
In the step-by-step solution provided, solving for the height involves substituting the given angle \(78.87^{\circ}\) and adjacent side (200 feet) into the tangent formula:

• \( \tan(78.87^{\circ}) = \frac{x}{200} \)
• "x" is the height of the sign.

By rearranging the formula to solve for \(x\), you multiply both sides by the adjacent side:
\( x = 200 \times \tan(78.87^{\circ}) \)
Once you calculate \( \tan(78.87^{\circ}) \), you find it approximately equals 4.873. Thus, multiplying the tangent value by the distance yields the height of approximately 974.6 feet. Understanding these steps allows one to confidently determine heights, a practical skill in various fields where direct measurement isn't possible.