Problem 58 - Algebra and Trigonometry

Question

Height of a Balloon 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 3.64 miles high.

Step 1: Understand the Problem

The problem is about finding the height of a balloon using angles of depression from two mileposts. We have two angles of depression: \( 20^{\circ} \) and \( 22^{\circ} \). The balloon is directly above a line on the ground, which is assumed to be straight.

Step 2: Visualize the Scenario

Imagine a triangle formed by the balloon's height, the horizontal distance on the road between the mileposts, and the sightline to each milepost. The angles of depression can be thought of as angles of elevation from the mileposts to the balloon. Thus, we have two right triangles with common vertex at the balloon.

Step 3: Apply Trigonometry

For each triangle, we use the tangent function of the angles to relate the height of the balloon ( h ) to the horizontal distance ( d or d+1 ) from where the balloon is directly overhead. The tangent of the angle is the opposite side (height h ) over the adjacent side (distance d ). First triangle: \( \tan(22^{\circ}) = \frac{h}{d} \). Second triangle: \( \tan(20^{\circ}) = \frac{h}{d+1} \).

Step 4: Set Up Equations

From the tangent equations: \( h = d \cdot \tan(22^{\circ}) \) and \( h = (d+1) \cdot \tan(20^{\circ}) \). Since both expressions equal h , set them equal: \( d \cdot \tan(22^{\circ}) = (d+1) \cdot \tan(20^{\circ}) \).

Step 5: Solve for d

Rearrange the equation: \( d \cdot \tan(22^{\circ}) = d \cdot \tan(20^{\circ}) + \tan(20^{\circ}) \). Solve for d : \( d = \frac{\tan(20^{\circ})}{\tan(22^{\circ}) - \tan(20^{\circ})} \).

Step 6: Calculate d and h

Calculate d using the tangent values, then substitute d back into any of the tangent equations to find h , the height of the balloon. With tangent values \( \tan(22^{\circ}) \approx 0.404 \) and \( \tan(20^{\circ}) \approx 0.364 \), find \( d \approx 9.0 \) and thus \( h = d \cdot \tan(22^{\circ}) \approx 9.0 \cdot 0.404 \approx 3.64 \) miles.

Key Concepts

Understanding Angles of Depression

Angles of depression are an interesting concept in trigonometry used to determine the distance between an observer and an object located below the horizontal level of the observer. When looking at an object below, the angle formed between a horizontal line from the observer's eye level and the line of sight to the object is called the angle of depression.
Commonly found in problems involving height and distance, this concept helps us determine how far away something is without physically measuring the distance.

• It is important to remember that an angle of depression from a point, say a balloon, to an object on the ground can also be viewed as an angle of elevation from the object to the balloon. This duality helps in applying trigonometric rules easily.
• In the exercise, the balloonists measured angles of depression from the balloon to mileposts on the ground, helping us calculate the balloon's height above the ground.

Analyzing Right Triangles

Right triangles are a fundamental part of trigonometry, featuring one 90-degree angle. They serve as a foundational element in solving numerous trigonometric problems, including those involving angles of depression.
Understanding the properties and relationships within right triangles allows us to use trigonometric functions like sine, cosine, and tangent effectively.

• In the practical scenario of measuring the balloon's height, two right triangles are formed. Each triangle shares a common vertex at the balloon to the road's endpoints, where angles of depression were measured.
• The height of the balloon serves as the opposite side of these triangles, while the horizontal distance along the road functions as the adjacent side.

The known angles of depression can be utilized as angles of elevation from the road to the balloon. From this information, we can apply trigonometry to find unknown lengths, such as the balloon's height.

Utilizing the Tangent Function

The tangent function is a trigonometric function that relates the angles and sides of a right triangle. It is defined as the ratio of the opposite side to the adjacent side. For an angle situated in the triangle, if you denote it as \( \theta \), then \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
The tangent function is especially useful when dealing with problems involving heights and distances, like determining how high a balloon floats.

• In our exercise, we use the tangent function to relate the height of the balloon (opposite side) and the horizontal distances (adjacent sides) from the mileposts to the balloon's vertical projection on the ground.
• Using the angles of depression and recognizing them as angles of elevation, we set up equations using the tangent function. This allowed us to solve for unknown lengths - here, the height of the balloon.
• By finding tangent values for the angles given, we solve for the horizontal distance first and then use it to find out how high the balloon is.