A 25 feet high lighthouse stands on the side of a hill. At a point of 80 feet from the foot of the lighthouse, measured straight down the hill, the lighthouse subtends an angle of 17 degrees. What angle does the side of the hill make with the lighthouse? A. 44 degrees 52’ B. 69 degrees 19’ C. 70 degrees 8’ D. 93 degrees 41’

Let's analyze the problem and solve it step by step.

Given:
- Height of the lighthouse, \( h = 25 \) feet.
- Distance from the foot of the lighthouse to a point on the hill, \( d = 80 \) feet.
- Angle subtended by the lighthouse at that point, \( \theta = 17^\circ \).

We need to find the angle \( \alpha \) that the side of the hill makes with the lighthouse.

### Step 1: Understanding the Geometry
Consider a right triangle where:
- The height of the lighthouse \( h = 25 \) feet is the vertical side.
- The distance \( d = 80 \) feet is along the hill, forming the hypotenuse of the triangle.
- The angle \( \theta = 17^\circ \) is the angle subtended by the lighthouse from that point.

### Step 2: Finding the Angle of Elevation
Using the formula for the tangent in a right triangle:
\[
\tan(\theta) = \frac{h}{x}
\]
Where:
- \( \theta = 17^\circ \)
- \( h = 25 \) feet
- \( x \) is the horizontal distance from the foot of the lighthouse to the point on the hill.

So,
\[
\tan(17^\circ) = \frac{25}{x}
\]

\[
x = \frac{25}{\tan(17^\circ)}
\]

Now, calculate \( x \):
\[
x \approx \frac{25}{0.3057} \approx 81.76 \text{ feet}
\]

### Step 3: Finding the Angle \( \alpha \)
The side of the hill forms the hypotenuse \( d \) in the triangle. To find the angle \( \alpha \) between the hill and the horizontal plane, we use:

\[
\cos(\alpha) = \frac

Learn how to solve the angle of elevation problem involving a lighthouse on a hill. This problem uses trigonometric concepts such as tangent and cosine functions in right triangles. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation