Problem 24 - Precalculus: Functions and Graphs

Question

The peak of Mt. Fuji in Japan is approximately \(12,400\) feet high. A trigonometry student, several miles away, notes that the angle between level ground and the peak is \(30^{\circ} .\) Estimate the distance from the student to the point on level ground directly beneath the peak.

Step-by-Step Solution

Answer

The student is approximately 4.072 miles from the base of Mt. Fuji.

Step 1: Understand the Scenario

In this scenario, the student is standing a certain distance away from Mt. Fuji. They see the peak of the mountain at an angle of elevation of \(30^{\circ}\). We need to find out the horizontal distance between the student and the base of Mt. Fuji.

Step 2: Identify the Trigonometric Relationship

The situation involves a right triangle where the angle of elevation is \(30^{\circ}\), the opposite side is the height of Mt. Fuji (12,400 feet), and the adjacent side is the distance on level ground (which we need to find). The tangent function relates the opposite side to the adjacent side in a right triangle.

Step 3: Set Up the Tangent Equation

Using the tangent function: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), we have: \[ \tan(30^{\circ}) = \frac{12,400}{d} \]Where \(d\) is the distance from the student to the point directly beneath the peak.

Step 4: Solve for the Distance

First, calculate \( \tan(30^{\circ}) \), which is approximately \(0.577\). Then, rearrange the equation to solve for \(d\):\[ d = \frac{12,400}{0.577} \]Evaluating this gives:\[ d \approx 21,493 \text{ feet} \]

Step 5: Convert Feet to Miles (if necessary)

Since the problem states that the student is several miles away, it is more practical to express this distance in miles. Knowing that 1 mile equals 5,280 feet, we convert the distance:\[ \text{Distance in miles} = \frac{21,493}{5,280} \approx 4.072 \text{ miles} \]

Key Concepts

Angle of Elevation

The angle of elevation is a critical concept in trigonometry. It describes the angle between the horizontal plane and the line of sight to an object above the horizontal, such as the peak of a mountain. Imagine standing on the ground and looking up at an object like a towering building or a mountain peak. The angle your line of sight makes with the ground is the angle of elevation.

• It is measured in degrees.
• This angle is always above the horizontal line.

In our scenario with Mt. Fuji, the student observes the peak at an angle of elevation of 30°. Understanding this concept helps us relate different parts of a geometric figure, like a right triangle, to solve for unknown distances.

Tangent Function

The tangent function is one of the primary trigonometric functions and is especially useful in right triangle problems. It relates the angle of a right triangle to the lengths of its opposite side and adjacent side. Defined as the ratio of these two sides, it facilitates solving for unknown dimensions.

• Formula: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• \( \theta \) is the angle of interest.

Using the tangent function in our problem, we find the equation \( \tan(30°) = \frac{12,400}{d} \), where 12,400 feet is the height of Mt. Fuji. Solving this equation helps us find the horizontal distance to the mountain's base.

Right Triangle

A right triangle is a triangle with one angle measuring exactly 90°. This type of triangle is fundamental in trigonometry and forms the basis for understanding how the tangent function works.

• The sides of a right triangle are categorized as opposite, adjacent, and hypotenuse.
• The right angle is always positioned between the two legs (adjacent and opposite).

In the Mt. Fuji problem, the right triangle is formed by:

• The peak's height as the opposite side.
• The ground distance as the adjacent side.
• An implied hypotenuse, running from the observer to the peak.

Using this structure, we apply trigonometric ratios to find unknown distances, which in this case, is the student's distance from the mountain.

Distance Conversion

Converting units of distance is often necessary when interpreting results in real-world contexts. In many problems, measurements are given in one unit, like feet, but are more practical in another unit, such as miles.

• 1 mile is equal to 5,280 feet.
• For accurate conversions, use the ratio of these units.

After calculating the distance from the student to the mountain base as approximately 21,493 feet, we convert it into miles. By dividing by 5,280, we find this corresponds roughly to 4.072 miles. These conversions ensure that results are easy to interpret, especially when considering distances that people often measure in miles.