Problem 79 - Precalculus

Question

The angle of inclination of a mountain with triple black diamond ski trails is \(65^{\circ}\) If a skier at the top of the mountain is at an elevation of 4000 feet, how long is the ski run from the top to the base of the mountain? Round to the nearest foot.

Step-by-Step Solution

Answer

The ski run is approximately 4413 feet long.

Step 1: Understanding the Problem

We are asked to find the length of the ski run from the top to the base of the mountain given the angle of inclination and the elevation.

Step 2: Identifying Known Values

The angle of inclination of the mountain given is \(65^{\circ}\), and the vertical elevation of the skier at the top of the mountain is 4000 feet.

Step 3: Using Trigonometric Relations

The vertical elevation is opposite to the angle of inclination. We need to find the hypotenuse of the right triangle formed, which represents the length of the ski run.

Step 4: Applying the Sine Function

We use the sine function, \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \theta = 65^{\circ} \), and the opposite side is 4000 feet. Hence, \( \sin(65^{\circ}) = \frac{4000}{\text{hypotenuse}} \).

Step 5: Solving for Hypotenuse

Rearrange the formula to solve for the hypotenuse: \( \text{hypotenuse} = \frac{4000}{\sin(65^{\circ})} \).

Step 6: Calculating the Hypotenuse

Calculate \( \sin(65^{\circ}) \approx 0.9063 \), then compute the hypotenuse: \( \text{hypotenuse} = \frac{4000}{0.9063} \approx 4413 \) feet.

Step 7: Final Answer

The length of the ski run from the top to the base of the mountain is approximately 4413 feet.

Key Concepts

Angle of Inclination

The angle of inclination is a fundamental concept in geometry and trigonometry. It refers to the angle formed between the horizontal plane and an inclined surface or line. In the context of our exercise, the mountain's angle of inclination is given as \(65^{\circ}\). This measures how steep the mountain is from the base to the top. A higher angle generally indicates a steeper slope.

Understanding the angle of inclination helps us solve various problems involving right triangles. For instance, in a right triangle, the angle of inclination helps us find relationships between different sides using trigonometric functions.

Key aspects include:

• Angle of elevation: Observed from a point below the object.
• Angle of depression: Observed from a point above the object.
• Helps in determining heights and distances in trigonometry.

Mastering the concept of the angle of inclination can significantly aid in solving more complex geometrical problems.

Right Triangle

A right triangle is a type of triangle that has a 90-degree angle. It is one of the most well-studied geometric shapes because of its unique properties:

The right triangle features two legs and a hypotenuse. The hypotenuse is always the longest side and is opposite the right angle. The other two sides, known as legs, are the base and height of the triangle.

Right triangles play a crucial role in trigonometry:

• They allow the use of trigonometric ratios (sine, cosine, tangent).
• They are associated with Pythagorean theorem applications.

In the given exercise, the right triangle is formed by:

• The vertical elevation (4000 feet) as one leg.
• The hypotenuse (ski run length) as the side we want to determine.

Right triangles can act as a powerful tool for solving real-world problems involving angles and distances.

Sine Function

The sine function is one of the primary trigonometric functions used to relate the angles and sides of right triangles. It is denoted as \( \sin(\theta) \), where \( \theta \) is an angle in a right triangle.

The basic definition states that the sine of an angle is the ratio of the length of the opposite side to the hypotenuse:

• Formula: \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

This function is vital for calculating unknown sides or angles in right triangles.

Steps to use the sine function in the exercise:

• Identify \( 65^{\circ} \) as \( \theta \), the angle of inclination.
• The opposite side is the mountain's elevation (4000 feet).
• Use the sine function to find the hypotenuse length, or the ski run.

By rearranging the formula \( \sin(65^{\circ}) = \frac{4000}{\text{Hypotenuse}} \), we derive that \( \text{Hypotenuse} = \frac{4000}{\sin(65^{\circ})} \).

The calculated hypotenuse length of approximately 4413 feet gives us the ski run length, showcasing the practical application of the sine function.