Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

As a hot-air balloon rises vertically, its angle of elevation from a point \(P\) on level ground 110 kilometers from the point \(Q\) directly underneath the balloon changes from \(19^{\circ} 20^{\prime}\) to \(31^{\circ} 50^{\prime}\) (see the figure). Approximately how far does the balloon rise during this period?

Step-by-Step Solution

Answer

The balloon rises approximately 29.788 kilometers.

Step 1: Understand the Problem

We need to find out how much the balloon rises as it travels vertically in the sky. This can be calculated by determining the change in altitude between the two angles of elevation from point P on the ground.

Step 2: Convert Angles to Decimal Degrees

First, convert the given angles from degrees and minutes to decimal degrees: \(19^{\circ} 20^{\prime} = 19 + \frac{20}{60} = 19.3333^{\circ}\) and \(31^{\circ} 50^{\prime} = 31 + \frac{50}{60} = 31.8333^{\circ}\).

Step 3: Define Trigonometric Relationships

Using the right triangle formed by the ground and vertical height of the balloon, we use tangent function which relates angle of elevation to the opposite side (height) and adjacent side (distance from point Q). The tangent of an angle is \( \tan(\theta) = \frac{h}{d}\) where \( h \) is height, and \( d \) is distance from P to Q, here 110 km.

Step 4: Calculate Initial Height

Calculate height when angle is \(19.3333^{\circ}\): \[ h_1 = 110 \times \tan(19.3333^{\circ}) \] Compute \( \tan(19.3333) \approx 0.3514 \), so: \[ h_1 = 110 \times 0.3514 = 38.654 \text{ km} \]

Step 5: Calculate Final Height

Calculate height when angle is \(31.8333^{\circ}\): \[ h_2 = 110 \times \tan(31.8333^{\circ}) \] Compute \( \tan(31.8333) \approx 0.6222 \), so: \[ h_2 = 110 \times 0.6222 = 68.442 \text{ km} \]

Step 6: Determine the Change in Height

Finally, find the difference between the final and initial heights, which gives the rise of the balloon: \[ \Delta h = h_2 - h_1 = 68.442 - 38.654 = 29.788 \text{ km} \]

Key Concepts

angle of elevation

The concept of an angle of elevation is all about looking up from a horizontal plane to an object above. Imagine standing on the ground and looking up at a hot air balloon. The angle your line of sight makes with the ground is the angle of elevation. It's like tilting your head upwards to see something elevated. In most trigonometry problems, this angle helps us find distances and heights.

The angle of elevation changes based on where you stand relative to the object. If you walk closer to the object, the angle increases, and if you walk away, it decreases. Therefore, understanding this concept is crucial when solving problems involving heights, like determining how far a balloon rises, based on its elevation angle from the ground level point.

tangent function

The tangent function is a fundamental concept in trigonometry. It relates the angle of elevation to two sides in a right triangle: the opposite side and the adjacent side. Specifically, it is defined as the ratio of the length of the opposite side to that of the adjacent side. In mathematical terms, it's expressed as \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \].

In our balloon problem, the opposite side is the vertical height of the balloon, and the adjacent side is the horizontal distance from the observer on the ground to the point directly below the balloon. This relationship makes the tangent function incredibly useful for calculating unknown heights or distances, especially when only the other two elements are known. By knowing the distance and angles of elevation, you can determine how much the balloon has risen using this trigonometric ratio.

right triangle

In trigonometry, a right triangle is a triangle that has one angle of exactly 90 degrees. The right triangle is pivotal in trigonometry as it provides a framework for defining trigonometric functions. In our problem involving the hot-air balloon, the right triangle is formed by: - The vertical rise (height) of the balloon - The horizontal distance from the observer to the point beneath the balloon - The line of sight from the observer to the balloon.

The right angle is typically at the ground level, where the vertical line meets the horizontal distance. The angle of elevation is one of the other two angles of the triangle. This triangular setup allows us to utilize trigonometric functions like tangent to solve for unknown variables by exploiting the set relationships between angles and sides in right triangles.

vertical height

Vertical height refers to the measure of how tall or how high an object stands from a reference point, usually the ground or base. In the context of a hot-air balloon, the vertical height is the distance it rises in the sky above the ground level.

This distance can be tricky to measure directly, particularly when the object is far overhead. Instead, we often rely on trigonometric relationships. Using angles of elevation and known distances from an established point, we can apply the tangent function to determine this height. This method involves forming a right triangle as discussed earlier, where the vertical height forms the opposite side of the triangle. As the balloon ascends, calculating the change in this height becomes straightforward, given the angles change due to the balloon’s movement.