Problem 41 - A Graphical Approach to Precalculus with Limits

Question

An antenna is on top of the center of a house. From a point on the ground 28.0 meters from the center of the house, the angle of elevation to the top of the antenna is \(27^{\circ} 10^{\prime},\) and the angle of elevation to the bottom of the antenna is \(18^{\circ} 10^{\prime} .\) Find the height of the antenna.

Step-by-Step Solution

Answer

The antenna is approximately 5.3 meters tall.

Step 1: Understand the Problem

To find the height of the antenna, we need to determine the heights associated with two angles of elevation from the same point on the ground to different vertical positions: the top and bottom of the antenna. These are essentially two separate right-angled triangles sharing a common horizontal distance.

Step 2: Convert Angles from Degrees and Minutes

Convert the angles of elevation from degrees and minutes to decimal degrees. For \(27^{\circ} 10^{\prime}\), we have \(27.1667\) degrees (since \(10^{\prime} = 10/60 = 0.1667\)), and for \(18^{\circ} 10^{\prime}\), we have \(18.1667\) degrees.

Step 3: Calculate the Height to the Top of the Antenna

Using the angle of elevation to the top of the antenna (\(27.1667^{\circ}\)), the height to the top of the antenna \(h_1\) can be found by the tangent function: \(\tan(27.1667^{\circ}) = \frac{h_1}{28}\). Solving for \(h_1\) gives: \[h_1 = 28 \times \tan(27.1667^{\circ})\]\[h_1 \approx 14.4 \text{ meters}\].

Step 4: Calculate the Height to the Base of the Antenna

Using the angle of elevation to the bottom of the antenna (\(18.1667^{\circ}\)), calculate the height to the base of the antenna \(h_2\) with: \(\tan(18.1667^{\circ}) = \frac{h_2}{28}\).\[h_2 = 28 \times \tan(18.1667^{\circ})\]\[h_2 \approx 9.1 \text{ meters}\].

Step 5: Find the Height of the Antenna

The height of the antenna alone can be found by subtracting the height to the base of the antenna from the height to the top of the antenna:\[h = h_1 - h_2\]\[h = 14.4 - 9.1\]\[h \approx 5.3 \text{ meters}\].

Key Concepts

Angles of Elevation

When you hear the term "angle of elevation," imagine looking up at something taller than yourself, like a tall building or a mountain top. This angle is made between the line of sight (your eye line straight to the object) and the horizontal line from your eye level. It's a handy concept when you're dealing with objects viewed from below their height.

In real-world scenarios, angles of elevation are crucial to determine how tall an object is without directly measuring it. Imagine standing at a known distance from a tower and using simple tools or even apps to find out how far you've tilted your head to see the top.

This angle, along with the distance to the object, allows you to form a right-angled triangle to calculate the height of the object above your eye level using trigonometric functions such as tangent.

Right-Angled Triangles

A right-angled triangle makes understanding elevation angles easier. These triangles have one angle of exactly 90 degrees, known as the right angle.

In the scenario where you are calculating the height of an antenna, the ground and the line of your sight to either the top or bottom of the antenna form the two shorter sides of the triangle; these are the base and the height.

• The hypotenuse is the line created by your line of sight.
• The base is the solid ground distance to the object's base.
• The height represents the object’s elevation you wish to calculate.

The properties of these triangles are perfect for calculations as they allow the use of trigonometric functions efficiently.

Tangent Function

The tangent function is a primary tool in trigonometry to find an unknown side or angle in right-angled triangles. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.

Mathematically, it's represented as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). In our antenna problem:

• "Opposite" refers to the height of the antenna, which we need to calculate.
• "Adjacent" is the horizontal distance from the observer to the base of the antenna.

With the known angles and the distance (adjacent side), you can rearrange the formula to solve for the "opposite" side, or height, providing a precise result, even when a direct measure isn't feasible.