Problem 107 - Algebra and Trigonometry

Question

A hot-air balloon is rising vertically. From a point on level ground 120 feet from the point directly under the passenger compartment, the angle of elevation to the balloon changes from \(37.1^{\circ}\) to \(62.4^{\circ} .\) How far, to the nearest tenth of a foot, does the balloon rise during this period?

Step-by-Step Solution

Answer

To get the rise in the balloon's height, calculate difference between the initial and final height using the tangent ratios.

Step 1: Understand the Given

The problem includes an initial elevation angle of \(37.1^{\circ}\) and a final angle of \(62.4^{\circ}\). The horizontal distance from the balloon to a point on the ground is 120 feet. This forms two right triangles, one for the initial position of the balloon and one for the final position.

Step 2: Apply Tangent Ratio for Initial Position

First, calculate the initial height of the balloon using the formula for the tangent of an angle in a right triangle: \(\tan(\theta) = \frac{opposite}{adjacent}\). For the initial balloon position, \(\theta = 37.1^{\circ}\), opposite is the initial height (unknown), and adjacent is 120 feet. We rearrange the formula to solve for the opposite side, giving \(Initial \ Height = 120 \cdot \tan(37.1^{\circ})\).

Step 3: Apply Tangent Ratio for Final Position

Similarly, calculate the final height of the balloon using the tangent ratio. This time, \(\theta = 62.4^{\circ}\), and adjacent remains 120 feet. We get \(Final \ Height = 120 \cdot \tan(62.4^{\circ})\).

Step 4: Calculate Height Difference

Subtract the initial height from the final height to calculate how much the balloon rises between the two time points.

Key Concepts

Angle of Elevation

In trigonometry, the angle of elevation refers to the angle formed between the horizontal line of sight and the line of sight up to an object. This concept is crucial when dealing with problems that involve heights and distances you need to calculate, such as the height a hot air balloon reaches over time. By imagining yourself standing on level ground and looking up at an object, the angle you tilt your head is essentially the angle of elevation.

• The angle is always measured from the horizontal.
• It applies specifically when the point you're observing is higher than your observation point.
• Using this angle with trigonometric functions allows you to determine distances and heights accurately.

In our example with the hot-air balloon, this angle changes from a smaller angle (37.1°) to a larger angle (62.4°), suggesting that the balloon is rising.

Right Triangle

A right triangle is a polygon with three sides and one 90-degree angle. The right triangle is foundational in trigonometry because it allows us to apply trigonometric ratios to solve for unknown sides or angles. Each right triangle consists of:

• The hypotenuse, which is always opposite the right angle and is the longest side.
• The adjacent side, which is next to the angle of interest and forms a part of the horizontal line.
• The opposite side, which is across from the angle of interest and forms the vertical line in the right triangle.

When dealing with the hot-air balloon scenario, the scenario forms two distinct right triangles. One triangle corresponds to the balloon's initial position, and the other corresponds to its final position. Each of these triangles shares a common adjacent side (120 feet in this case), representing the horizontal distance from the point underneath the balloon to the observer's point on the ground.

Tangent Ratio

The tangent ratio is one of the primary trigonometric functions used to relate the angles and sides of right triangles. Specifically, for a given angle in a right triangle:

• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
• This formula helps us find unknown lengths in right triangles when an angle and one side length are known.
• In the balloon problem, we use the tangent ratio twice—first to find the initial height of the balloon, and second to find its final height.

For both the initial and final positions of the balloon, the tangent ratios are used in association with the known angles of elevation (37.1° and 62.4°) and the horizontal distance (120 feet) to calculate the actual heights.