https://file.aiquickdraw.com/mathbot/file/17285133283908aqca1ed.jpg

This problem can be solved using trigonometry, specifically applying the sine, cosine, or tangent functions.

### Problem Breakdown:
- Spencer is at the top of a mountain 1500 feet high.
- The angle of elevation from the bottom to the top of the mountain is 25°.
- We need to find the length of Spencer’s snowboarding run, which corresponds to the hypotenuse of a right triangle, where:
  - The altitude of the mountain (1500 feet) is the opposite side.
  - The angle of elevation is 25°.

### Solution Approach:
In this case, we can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta = 25^\circ\)
- Opposite side (altitude) = 1500 feet
- Hypotenuse (run length) = \(x\) (the unknown value)

Rearranging the equation to solve for the hypotenuse:
\[
x = \frac{\text{opposite}}{\sin(\theta)} = \frac{1500}{\sin(25^\circ)}
\]

### Calculation:
\[
x = \frac{1500}{\sin(25^\circ)} \approx \frac{1500}{0.4226} \approx 3549.3 \, \text{feet}
\]

So, the length of Spencer's run is approximately **3549.3 feet**.

Let me know if you need further details or have any questions.

Here are some related questions:
1. How would you find the length of the run if the angle of elevation were different, say 30°?
2. What is the importance of the sine function in solving problems like this?
3. How would the problem change if you were given the length of the run and asked to find the angle of elevation?
4. What is the relationship between angle of elevation and the steepness of a slope?
5. How does the cosine function apply in solving similar problems?

**Tip:** Always make sure your calculator is set to the correct mode (degrees or radians) based on the problem’s units when solving trigonometric problems.

Spencer is going snowboarding. He rode the ski lift to the top of the mountain, which is at an altitude of 1500 feet. The angle of elevation from the bottom to the top of the mountain is 25°. What is the length of Spencer's run (his trip down the hill)? Round your answer to the nearest tenth of a foot.

Learn how to solve a trigonometry problem to find the length of a snowboarding run. Spencer's trip down the hill is modeled as a right triangle, using an angle of elevation of 25° and a mountain altitude of 1500 feet. Detailed step-by-step solution provided, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation