Problem 69 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. From the basket of a hot-air balloon 100 feet above the ground, the angle of depression of a point \(A\) on the ground is \(10.5^{\circ} .\) What is the distance from point \(A\) to the basket of the balloon?

Step-by-Step Solution

Answer

The distance from point \(A\) to the basket of the balloon is approximately 536.3907 feet.

Step 1: Identifying the Variables

From the problem, the following variables can be identified: The height of the balloon from the ground \(h = 100\) feet, the angle of depression \(\alpha = 10.5^{\circ}\). The distance from point \(A\) to the basket of the balloon is denoted \(d\).

Step 2: Set Up the Right Triangle and Identify the Relevant Trigonometric Identity

In any right triangle, the tangent of an angle is the ratio of the side opposite the angle to the side adjacent to the angle. In this case, the angle of interest is \(\alpha\), the height of the balloon is the side opposite \(\alpha\) and the distance \(d\) is the side adjacent to \(\alpha\). Therefore, the relevant trigonometric identity is \(\tan (\alpha) = \frac{h}{d}\).

Step 3: Insert Known Values and Solve for Unknown

Substitute height \(h = 100\) feet and angle \(\alpha = 10.5^{\circ}\) into the equation \(\tan (\alpha) = \frac{h}{d}\). This gives \(\tan (10.5^{\circ}) = \frac{100}{d}\). Solving for \(d\), we find \(d = \frac{100}{\tan (10.5^{\circ})} \approx 536.3907\) feet when rounded to four decimal places. So, the distance from point \(A\) to the balloon is approximately 536.3907 feet.

Key Concepts

Understanding Angle of Depression

The angle of depression is a concept often used in real-world trigonometry problems, particularly those involving right triangles. Imagine yourself at the top of a tall building, looking down at a distant object on the ground like a car. The angle between your line of sight and the horizontal line (parallel to the ground) is what we call the angle of depression.

• The angle is measured from the horizontal line down to the line of sight.
• This can also be compared with the angle of elevation, which is measured upwards from the ground level.

In the balloon problem, the basket is 100 feet above the ground, and the point on the ground is considered to form this angle of depression, which is given as 10.5°. By understanding this angle, we can start tackling the math using trigonometric functions to find unknown sides of the triangle.

Exploring the Tangent Function

When dealing with right triangles, the tangent function is especially useful because it relates two particular sides of the triangle through the angle. The tangent of an angle in a right triangle (often denoted as \(\tan \theta\)) is defined as the ratio of the opposite side to the adjacent side.
In mathematical terms, \(\tan (\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

• 'Opposite' refers to the side opposite the angle you are examining, which in our problem is the height of the balloon (100 feet).
• 'Adjacent' is the side next to the angle, which we need to find—the distance from the point on the ground to the basket.

By using the formula \(\tan (10.5^{\circ}) = \frac{100}{d}\), where \(d\) is the distance we need to discover, we can rearrange to solve for \(d\). By calculating, you uncover that \(d \approx 536.3907\) feet.

Solving Real-World Application Problems

Real-world application problems like the hot-air balloon scenario are perfect for demonstrating the practical use of mathematical concepts such as trigonometry. Here’s how these problems unfold:

• First, identify what you know: the height, the angle, and what you need to find—which is typically a length or distance.
• Next, apply relevant trigonometric functions like tangent, sine, or cosine, based on the parts of the triangle you’re dealing with.
• Finally, solve for the unknown by substituting the known values in the trigonometric equation you've set up.

In our example, the relationship between the height and the horizontal distance is simplified by using the tangent function. This approach is not only efficient but also logical, aligning perfectly with many real-life situations where angles and heights are involved. Such exercises strengthen problem-solving skills and a practical understanding of mathematics.