Problem 305 - Precalculus

Question

There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be \(36^{\circ} .\) From the same location, the angle of elevation to the top of the lightning rod is measured to be \(38^{\circ} .\) Find the height of the lighthning rod.

Step-by-Step Solution

Answer

The height of the lightning rod is approximately 25.22 feet.

Step 1: Understand the Problem

You are given two angles of elevation from a point 500 feet away from the base of a building. The first angle of elevation to the top of the building is \(36^{\circ}\), and the second angle of elevation to the top of the lightning rod is \(38^{\circ}\). The task is to find the height of the lightning rod.

Step 2: Use the Tangent Function for Height Calculation

To find the heights, we use the tangent of an angle, which is the ratio of the opposite side to the adjacent side in a right triangle. For the top of the building:\[tan(36^{\circ}) = \frac{h_1}{500}\]For the top of the lightning rod:\[tan(38^{\circ}) = \frac{h_2}{500}\]where \(h_1\) is the height of the building and \(h_2\) is the total height including the lightning rod.

Step 3: Solve for Heights

Rearrange the equations to solve for \(h_1\) and \(h_2\):\[h_1 = 500 \cdot \tan(36^{\circ})\]\[h_2 = 500 \cdot \tan(38^{\circ})\]Calculate these using a calculator to find the heights.

Step 4: Calculate the Height of the Lightning Rod

The height of the lightning rod \(h_r\) is the difference between the total height \(h_2\) and the height of the building \(h_1\):\[h_r = h_2 - h_1\]Substitute the values of \(h_2\) and \(h_1\) to find \(h_r\).

Step 5: Solve with Calculated Values

Calculate \(h_1\) and \(h_2\): \[h_1 = 500 \cdot \tan(36^{\circ}) \approx 363.97 \text{ feet}\]\[h_2 = 500 \cdot \tan(38^{\circ}) \approx 389.19 \text{ feet}\]Now, find \(h_r\):\[h_r = 389.19 - 363.97 \approx 25.22 \text{ feet}\]Thus, the height of the lightning rod is approximately 25.22 feet.

Key Concepts

Angle of Elevation

The angle of elevation is a fascinating and practical concept in trigonometry. It describes the angle between the horizontal plane and the line of sight when an observer looks up at an object. This is typically measured from the observer's eye level to the top point of the object they're looking at. For instance, imagine standing at some distance from a towering building and observing its roof. Your line of sight, or vision line, would make a certain angle with a horizontal line going straight from your eyes. This angle is known as the angle of elevation.

To grasp it visually, picture a large triangle formed by three points: the observer's eye, the horizontal line from the observer to the bottom of the building, and the line of sight to the top of the building. The angle of elevation is part of this triangle and is a crucial aspect in solving problems involving heights and distances.

• It's always measured upwards from the horizontal.
• It's useful in a variety of fields, from navigation to architecture.

By understanding angles of elevation, you can calculate the heights of inaccessible objects using only simple trigonometry.

Tangent Function

The tangent function is a fundamental concept in trigonometry, commonly used in situations involving right triangles. In simple terms, it describes the ratio of two sides of a right triangle: the opposite side to the adjacent side. This can be expressed as: \[\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\] This relationship helps you calculate an unknown side, provided you have one side's length and know the angle.

Returning to our example with the building and the lightning rod, the tan function lets you find the heights based on the angle of elevation. You can rearrange the basic tangent formula to find an unknown height (let's call it \( h \)) given a distance: \[h = \text{Distance} \times \tan(\text{Angle of elevation})\] This is exactly how we deduced the heights in the textbook solution. The tangent function not only works for finding heights but is equally useful for a variety of practical calculations.

• It is a part of the core set of trigonometric functions.
• Every angle in a right triangle has a specific tangent value, accessible in scientific calculators.

Right Triangle

The right triangle is undoubtedly a cornerstone of trigonometry. Defined as a triangle with one 90-degree angle, it forms the basis of many mathematical principles. Right triangles not only feature this right angle but also have two other angles whose sum will always equal 90 degrees (because the total for any triangle is 180 degrees).

When you apply trigonometric concepts like tangent, sine, and cosine, it's often within the context of these triangles. The three sides of a right triangle are known as the hypotenuse, the adjacent side, and the opposite side relative to a specific angle. In problems like the one involving the lightning rod,

• The hypotenuse is the longest side opposite the right angle.
• The adjacent side is next to the angle in question.
• The opposite side is directly opposite the angle in question.

Understanding the dynamics of a right triangle lets you decode the heights and distances involved in real-world problems. Much of trigonometry boils down to knowing how to manipulate these three sides, presenting solutions through relationships like Pythagorean theorem and trigonometric functions. Mastery of the right triangle simplifies many complex scenarios, making them straightforward and solvable.