Problem 107 - Precalculus Essentials

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? (Section 4.8, Example 4 )

Step-by-Step Solution

Answer

The balloon rises \(\Delta h\) feet, which is calculated by subtracting the initial height from the final height.

Step 1: Identify the known values and what needs to be found

The distance from the observer to the point directly beneath the balloon is \(120\) ft and remains the same. The angles of elevation at two moments are given as \(37.1^{\circ}\) and \(62.4^{\circ}\) respectively. We are asked to find the distance that the balloon rises, between the two moments of time.

Step 2: Apply the tangent definition

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the tangent of the angle of elevation is the ratio of the height of the balloon to the distance from the observer to the point directly underneath the balloon. We can write this as: \[ \tan(\text{{angle}}) = \frac{{\text{{height}}}}{{120}} \]Solving this for the height gives:\[ \text{{height}} = 120 \times \tan(\text{{angle}}) \]

Step 3: Find the initial and final heights

Using the formula from step 2, we can find the height of the balloon at the two moments. For the initial height \(h_1\) , we use the initial angle \(37.1^{\circ}\):\[ h_1 = 120 \times \tan(37.1^{\circ}) \]And for the final height \(h_2\), we use the final angle \(62.4^{\circ}\):\[ h_2 = 120 \times \tan(62.4^{\circ}) \]

Step 4: Calculate the difference in heights

The difference in heights will give us how far the balloon has risen. We can find this difference by substracting the initial height from the final height:\[ \Delta h = h_2 - h_1 \]

Key Concepts

Right Triangle

In trigonometry, a right triangle is a triangle that has one angle measuring exactly 90 degrees. It consists of two legs and a hypotenuse, which is the side opposite the right angle and is also the longest side. Right triangles are a foundational concept in trigonometry and are used to define the trigonometric functions such as sine, cosine, and tangent.

In the context of our hot-air balloon problem, the right triangle formed includes:

• The height of the balloon, which changes and serves as the opposite side in the context of the tangent function.
• The distance from the observer to the point right below the balloon, which is 120 feet, acting as the adjacent side.

A right triangle allows us to use the properties of trigonometric functions to relate angles to side lengths. Thus, understanding the components and relationships within a right triangle facilitates solving problems involving elevation and height.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal ground and the line of sight to an object above the horizontal. It is always measured from the ground up, making it greater than zero and typically measured in degrees.

In our scenario with the rising hot-air balloon, the angle of elevation is critical to determining how much higher the balloon has ascended. As an observer shifts their gaze upwards from the horizontal level to track the balloon's ascent, this angle increases.

In practice, observing changes in angles of elevation involves:

• Using instruments such as a theodolite for precision, although simple estimations can be made with basic geometric principles.
• Tracking these angles at different times to find changes in height.

When solving such problems, understanding that the angle of elevation relates directly to tangent function calculations helps in determining height changes efficiently.

Tangent Function

The tangent function is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle.Symbolically, this can be expressed as:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For tasks involving angles of elevation, the tangent function is often the go-to method for computations.When we want to find a height, like in the balloon example, we rearrange the formula:

\[ \text{height} = \text{adjacent} \times \tan(\text{angle}) \]
In solving our problem, using the tangent function allowed us to calculate the initial and final heights of the balloon by plugging in the known distances and angle measurements.This approach is efficient because:

• It involves only basic arithmetic operations once the angle's tangent has been computed.
• It can be applied to determine the height change at both small and large angles of elevation.

Overall, the tangent function is indispensable when dealing with right triangles and elevation-related problems in trigonometry.