The altitude of a mountain peak is measured as shown in the figure to the right. At an altitude of 14 comma 512 feet on a different​ mountain, the​ straight-line distance to the peak of Mountain A is 27.7862 miles and the​ peak's angle of elevation is thetaequals5.8800 degrees.
​(a) Approximate the height​ (in feet) of Mountain A.
​(b) In the actual​ measurement, Mountain A was over 100 mi away and the curvature of Earth had to be taken into account. Would the curvature of Earth make the peak appear taller or shorter than it actually​ is?
14,512 ft
27.7862 mi
Question content area bottom
Part 1
​(a) The height of Mountain A is approximately

enter your response here feet.

We can solve this problem by using basic trigonometry, specifically the sine of the angle of elevation. Here's the approach:

### Part (a) Approximate the height of Mountain A

Given:
- The angle of elevation to the peak of Mountain A, \( \theta = 5.8800^\circ \)
- The straight-line distance to the peak of Mountain A, \( d = 27.7862 \) miles
- The altitude of the observation point (on the other mountain) is 14,512 feet

1. **Convert the straight-line distance to feet**:
   Since 1 mile = 5280 feet, we convert the distance:
   \[
   d = 27.7862 \times 5280 = 146,597.136 \text{ feet}
   \]

2. **Use the sine function**:
   The sine of an angle in a right triangle is the ratio of the opposite side (the height of Mountain A relative to the observation point) to the hypotenuse (the straight-line distance to the peak). The equation is:
   \[
   \sin(\theta) = \frac{\text{height of Mountain A relative to the observation point}}{\text{straight-line distance}}
   \]

   Solving for the height of Mountain A relative to the observation point:
   \[
   \text{height of Mountain A relative to the observation point} = d \times \sin(\theta)
   \]
   \[
   \text{height of Mountain A relative to the observation point} = 146,597.136 \times \sin(5.8800^\circ)
   \]
   
   Let's calculate that.

It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can use the following method to calculate the height of Mountain A:

1. **Convert the distance from miles to feet**:  
   \[
   \text{Distance in feet} = 27.7862 \times 5280 = 146,597.136 \text{ feet}
   \]

2. **Use the sine function**:  
   \[
   \text{Height relative to observation point} = 146,597.136 \times \sin(5.8800^\circ)
   \]
   You can plug this into a calculator to find the height relative to the observation point, and then add the 14,512 feet altitude of the observation mountain to get the total height of Mountain A.

### Part (b) Earth Curvature

The curvature of the Earth would cause the peak of Mountain A to appear shorter than it actually is when viewed from a large distance. This is because the observer is seeing over the curved surface of the Earth, which lowers the apparent height of distant objects.

---

Let me know if you have any questions or want more details! 

### Related Questions:
1. What is the difference between straight-line distance and ground distance in elevation problems?
2. How does the curvature of the Earth affect long-distance elevation measurements?
3. Can we approximate the height of a peak without trigonometry?
4. Why is it important to account for Earth's curvature in geographic measurements?
5. What is the impact of refraction on the angle of elevation over long distances?

**Tip:** Always ensure units are consistent when calculating distances and angles.

The altitude of a mountain peak is measured as shown in the figure to the right. At an altitude of 14,512 feet on a different mountain, the straight-line distance to the peak of Mountain A is 27.7862 miles and the peak's angle of elevation is theta = 5.8800 degrees. (a) Approximate the height (in feet) of Mountain A. (b) In the actual measurement, Mountain A was over 100 miles away, and the curvature of the Earth had to be taken into account. Would the curvature of the Earth make the peak appear taller or shorter than it actually is?

Learn how to approximate the height of Mountain A using trigonometry, the sine function, and angle of elevation. This problem involves a 5.8800-degree angle of elevation and a straight-line distance of 27.7862 miles. Understand how Earth's curvature affects long-distance measurements. Suitable for high school students (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