Question

a ranger station at the peak of a 3,000ft. mountain sighted a single tower on top of another mountain at an angle of elevation of 20 degrees. if gps positioning shows the directs distance between peaks as 10,500 ft. what is the altitude of the tower?

Answer 1

To find the altitude of a tower on a neighboring mountain with a 20-degree angle of elevation and a 10,500 ft distance between peaks, use tan(20 degrees) = (altitude of the tower - 3,000 ft)/10,500 ft to calculate that the tower's altitude is approximately 6,822 ft.

Step-by-step explanation:

The student's question asks about finding the altitude of a tower on a neighboring mountain when given the angle of elevation and the distance between the two mountain peaks. To solve this, we apply trigonometric functions specifically, the tangent function that relates angles to the opposite and adjacent sides in a right triangle.

Here's the step-by-step method:

1. Identify the angle of elevation to the top of the tower, which is 20 degrees.
2. Use the known distance between the mountain peaks as the adjacent side of the right triangle, which is 10,500 ft.
3. Apply the tangent function: tan(20 degrees) = (altitude of the tower - 3,000 ft)/10,500 ft.
4. Rearrange the formula to solve for the altitude of the tower: altitude of the tower = tan(20 degrees) * 10,500 ft + 3,000 ft.
5. Calculate the value using a calculator equipped with trigonometric function capabilities.

Let's perform the calculation:

• tan(20 degrees) ≈ 0.364
• 0.364 * 10,500 ft ≈ 3,822 ft
• Altitude of the tower ≈ 3,822 ft + 3,000 ft
• Altitude of the tower ≈ 6,822 ft