Problem 304 - Precalculus

Question

There is an antenna on the top of a building. From a location 300 feet from the building, the angle of elevation to the top of the building is measured to be \(40^{\circ} .\) From the same location, the angle of elevation to the top of the antenna is measured to be \(43^{\circ} .\) Find the height of the antenna.

Step-by-Step Solution

Answer

The antenna is approximately 28.02 feet tall.

Step 1: Identify the Problem

We need to find the height of the antenna on top of the building. We have two angles of elevation and the horizontal distance between the location of measurement and the building.

Step 2: Define the Variables

Let \(h_b\) be the height of the building, and \(h_a\) be the height of the antenna on top of the building. The total height to the top of the antenna is \(h_b + h_a\).

Step 3: Use the Angle to Building Top

For the angle of elevation \(40^{\circ}\) to the top of the building, use the tangent function: \(\tan(40^{\circ}) = \frac{h_b}{300}\). Solve for \(h_b\): \[ h_b = 300 \cdot \tan(40^{\circ}) \]

Step 4: Use the Angle to Antenna Top

For the angle of elevation \(43^{\circ}\) to the top of the antenna, also use the tangent function: \(\tan(43^{\circ}) = \frac{h_b + h_a}{300}\). Solve for \(h_b + h_a\): \[ h_b + h_a = 300 \cdot \tan(43^{\circ}) \]

Step 5: Calculate Antenna Height

Substitute \(h_b\) from Step 3 into the equation from Step 4 to find \(h_a\):\[ h_a = (300 \cdot \tan(43^{\circ})) - (300 \cdot \tan(40^{\circ})) \]

Step 6: Solve the Equation

Calculate the values:\[ h_b = 300 \cdot \tan(40^{\circ}) \approx 300 \cdot 0.8391 \approx 251.73 \text{ feet} \]\[ h_b + h_a = 300 \cdot \tan(43^{\circ}) \approx 300 \cdot 0.9325 \approx 279.75 \text{ feet} \]Subtract \(h_b\) from \(h_b + h_a\):\[ h_a = 279.75 - 251.73 \approx 28.02 \text{ feet} \]

Step 7: Conclusion

The height of the antenna is approximately 28.02 feet.

Key Concepts

Angle of Elevation

In trigonometry, the angle of elevation is a critical concept. It is the angle formed by the line of sight when looking upwards from a horizontal line. Imagine standing on the ground and looking up at the top of a building; the angle your line of sight makes with the horizontal ground is the angle of elevation.
An angle of elevation can help us determine various heights when combined with known distances. In this exercise, two angles of elevation are used: one for the top of the building and another for the top of the antenna. Each angle plays a valuable role in calculating different heights using trigonometric functions, notably the tangent function.

• The angle of elevation helps in determining the height when the distance is known.
• Allows for precise measurements without needing direct measurement tools to the top of tall objects.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the fundamental trigonometric functions. It relates an angle of a right triangle to the ratio of the opposite side to the adjacent side.
This relationship is essential in solving problems involving heights and distances.
For example, in our exercise:

• The tangent of the angle of elevation to the building's top is \( \tan(40^{\circ}) = \frac{h_b}{300} \).
• Essentially, this means the angle helps us calculate the opposite side (height) when the adjacent side (distance) is known.
• Similarly, \( \tan(43^{\circ}) \) is used to determine the height up to the antenna's top.

The tangent function proves indispensable in trigonometry for converting angles and distances into useful height measurements without physically reaching the top.

Solving Trigonometric Equations

Solving trigonometric equations involves using mathematical functions to find unknown values.
In our example, both \( h_b \) and \( h_a + h_b \) are unknowns, and they are found by setting up equations using the tangent function.
The main steps are:

• Write the equation using the tangent function for each angle of elevation.
• Solve for \( h_b \) using the known distance and angle: \( \frac{h_b}{300} = \tan(40^{\circ}) \).
• Calculate the total height using \( h_b + h_a = 300 \cdot \tan(43^{\circ}) \).
• Substitute the value of \( h_b \) into the second equation to solve for \( h_a \).

These steps highlight how trigonometric equations can be systematically manipulated to find precise values.

Calculation of Height

Calculating height using trigonometry involves a strategic approach.
We use given angles and distances to derive values that are difficult to measure directly. Here's a breakdown:
In the original exercise, you calculate two heights:

• The height of the building (\( h_b \)): \( 300 \cdot \tan(40^{\circ}) \).
• The combined height up to the antenna's top (\( h_b + h_a \)): \( 300 \cdot \tan(43^{\circ}) \).
• Finally, by deducting \( h_b \) from \( h_b + h_a \), the antenna height (\( h_a \)) is isolated.

This method allows one to achieve accurate height measurement outcomes using mathematical calculations instead of physical measurement.

Problem-solving in Trigonometry

Problem-solving in trigonometry often involves a series of logical steps applied systematically.
Understanding and applying trigonometric concepts can solve real-world problems efficiently.
In this exercise:

• Identify what you're solving for — here, it's the antenna's height.
• Establish the known factors: angles and distances.
• Apply trigonometric functions to find unknown heights.
• Perform calculations clearly and check each step for accuracy.

Trigonometry can be seen as a toolkit for resolving practical issues involving angles and heights. By understanding the underlying principles, students can develop adept problem-solving skills applicable in fields like engineering and construction.