Problem 58 - Algebra and Trigonometry

Question

A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be \(20^{\circ}\) and \(22^{\circ} .\) How high is the balloon?

Step-by-Step Solution

Answer

The balloon is approximately 1.4 miles high.

Step 1: Understand the Problem

The problem involves finding the height of a hot-air balloon above the ground. The angles of depression to two consecutive mileposts are given as 20° and 22°. We need to use these angles to determine the vertical height of the balloon.

Step 2: Sketch and Assign Variables

Sketch a diagram showing the hot-air balloon, the road, and the two mileposts. Let the height of the balloon be denoted by \(h\). The horizontal distances from the point directly below the balloon to the two mileposts will be denoted by \(x\) and \(x+1\), respectively, as the distance between mileposts is 1 mile.

Step 3: Trigonometric Relationships

Using the tangent function, which relates the angle of depression to the opposite and adjacent sides in a right triangle, we have:\[\tan(22^\ ext{°}) = \frac{h}{x}\]\[\tan(20^\text{°}) = \frac{h}{x+1}\]

Step 4: Express in Terms of Equations

From Step 3, set up the following equations:\[x = \frac{h}{\tan(22^\circ)}\]\[x + 1 = \frac{h}{\tan(20^\circ)}\]

Step 5: Solve for Height

Substitute the expression for \(x\) in one equation into the other:\[\frac{h}{\tan(22^\circ)} + 1 = \frac{h}{\tan(20^\circ)}\]Rearrange and solve for \(h\):\[h \left(\frac{1}{\tan(22^\circ)} - \frac{1}{\tan(20^\circ)}\right) = 1\]\[h = \frac{1}{\left(\frac{1}{\tan(22^\circ)} - \frac{1}{\tan(20^\circ)}\right)}\]

Step 6: Calculate Height

Calculate the height using the tangent values:\(\tan(22^\circ) \approx 0.4040 \textrm{ and } \tan(20^\circ) \approx 0.3640\)Substitute these into the equation:\[h \approx \frac{1}{\left(\frac{1}{0.4040} - \frac{1}{0.3640}\right)} \approx \frac{1}{(2.475 - 2.747)} \approx 1.4 \textrm{ miles}\]

Step 7: Conclusion

The height of the hot-air balloon is approximately 1.4 miles above the ground.

Key Concepts

Angle of Depression

An angle of depression is a key concept in trigonometry when analyzing situations involving lines of sight, such as this exercise with the hot-air balloon. Imagine standing at the point of view position, such as in the balloon. Looking straight out, you're looking at what is called the horizontal line. Now, if you look down towards the point on the ground, like a milepost, the angle formed between the horizontal line and your line of sight is the angle of depression.
This angle is measured from the horizontal downward to the object of interest, in this case, each milepost. In practical terms, if you're ever finding an angle of depression, you're comparing it to looking straight ahead and then angling your gaze lower.
In the exercise, there are two angles of depression, 20° and 22°, from the balloon to the mileposts. These angles help us determine distances and, consequently, the height of the balloon using trigonometric functions.

Tangent Function

In trigonometry, the tangent function is a critical tool that helps in solving problems involving angles and distances within right triangles. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side:

• Mathematically, this is represented as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• In this problem, \( \theta \) represents the angles of depression (20° and 22°), and using the tangent of these angles gives us the relationship between the height of the balloon (opposite side) and the horizontal distances (adjacent side) to the mileposts on the ground.

Practically, once you know two of these quantities—like the angle and the adjacent side—you can find the third, which is really beneficial in real-world navigation and mapping scenarios.
The exercise shows us how it can be used to express distances in equations and, finally, find the height of the balloon.

Right Triangle

Understanding the geometry of a right triangle is essential for solving the problem involving angles of depression. A right triangle is characterized by a right angle, which is one of its defining features. Relating to our balloon scenario:

• The height of the balloon represents one leg of the triangle, which is always perpendicular to the ground or the horizontal leg of the triangle.
• The horizontal distance from the point directly below the balloon to each milepost is the adjacent side.
• The line of sight from the balloon to the mileposts forms the hypotenuse of the triangle. However, since we are focusing on angles of depression and tangent functions, the hypotenuse is not directly used in our calculations here.

By effectively sketching and understanding the right triangle, you can identify these sides, use trigonometric functions, such as tangent, and solve for unknowns like the height of the balloon. This clear geometric understanding simplifies how we approach and resolve trigonometric problems.