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The angle of elevation to a mountaintop is 12°. If the mountaintop is directly above a point 19 miles away, what is the altitude of the mountaintop? The altitude of the mountaintop is three decimal places) miles. (Round your answer to

The angle of elevation to a mountaintop is 12°. If the mountaintop is directly above a point 19 miles away, what is the altitude of the mountaintop? The altitude of the mountaintop is three decimal places) miles. (Round your answer to

Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter67: Analysis Of Trigonometric Functions
Section: Chapter Questions
Problem 13A: Refer to the following figure in answering Exercises 7 through 13. It may be helpful to sketch...
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**Problem Statement:** The angle of elevation to a mountaintop is 12°. If the mountaintop is directly above a point 19 miles away, what is the altitude of the mountaintop? **Question:** The altitude of the mountaintop is __________ miles. (Round your answer to three decimal places) **Explanation:** To determine the altitude of the mountaintop, you can use trigonometric functions, specifically the tangent function, which relates the angle of elevation to the ratio of the altitude of the mountaintop and the horizontal distance. Here, the angle of elevation (θ) is 12°, and the horizontal distance (adjacent side, \(A\)) is 19 miles. Let \(H\) be the altitude of the mountaintop (opposite side). The tangent of the angle is given by: \[ \tan(\theta) = \frac{H}{A} \] Substituting the known values: \[ \tan(12°) = \frac{H}{19} \] Solving for \(H\): \[ H = 19 \times \tan(12°) \] Using a calculator to find \(\tan(12°)\): \[ \tan(12°) \approx 0.2126 \] Therefore: \[ H = 19 \times 0.2126 \approx 4.040 \] The altitude of the mountaintop is approximately 4.040 miles.
Transcribed Image Text:**Problem Statement:** The angle of elevation to a mountaintop is 12°. If the mountaintop is directly above a point 19 miles away, what is the altitude of the mountaintop? **Question:** The altitude of the mountaintop is __________ miles. (Round your answer to three decimal places) **Explanation:** To determine the altitude of the mountaintop, you can use trigonometric functions, specifically the tangent function, which relates the angle of elevation to the ratio of the altitude of the mountaintop and the horizontal distance. Here, the angle of elevation (θ) is 12°, and the horizontal distance (adjacent side, \(A\)) is 19 miles. Let \(H\) be the altitude of the mountaintop (opposite side). The tangent of the angle is given by: \[ \tan(\theta) = \frac{H}{A} \] Substituting the known values: \[ \tan(12°) = \frac{H}{19} \] Solving for \(H\): \[ H = 19 \times \tan(12°) \] Using a calculator to find \(\tan(12°)\): \[ \tan(12°) \approx 0.2126 \] Therefore: \[ H = 19 \times 0.2126 \approx 4.040 \] The altitude of the mountaintop is approximately 4.040 miles.
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