To solve this problem, let's break down the steps involved:
1. Understanding the Problem:
- We need to find the distance from a fishing boat to the base of a lighthouse.
- The angle of elevation from the boat to the top of the lighthouse is 5°.
- The height of the lighthouse above sea level is 161 feet.
2. Drawing the Picture:
- Sketch a right-angled triangle.
- Label the top of the lighthouse as point \(\( A \)\).
- Label the bottom of the lighthouse (sea level) as point \(\( B \)\).
- Label the position of the boat as point \(\( C \)\).
\(\[
\text{A}(Lighthouse)
\]\)
\(\[
|
\]\)
\(\[
|
\]\)
\(\[
|
\]\)
\(\[
| (Height = 161 \text{ feet})
\]\)
\(\[
|
\]\)
\(\[
|
\]\)
\(\[
B = \text{Base of lighthouse}
\]\)
\(\[
\_\_\_\_\_\_\_\_\_\_\_\_ \]\) \(\(\angle = 5^\circ \text{ (Angle of elevation)}\)\)
\(\[
C \text{ (Boat)}
\]\)
3. Setting up the problem:
- The height from the base to the top of the lighthouse (\(\(AB\)\)) is 161 feet (opposite side).
- The distance from the boat to the base of the lighthouse (\(\(BC\)\)) is the adjacent side.
- The angle of elevation (\(\(\angle BAC\)\)) is 5°.
4. Using Trigonometry:
- We will use the tangent function, which relates the opposite side to the adjacent side in a right-angled triangle:
\(\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]\)
- Here, \(\(\theta\)\) is 5°, the opposite side is 161 feet.
5. Formulating the Equation:
\(\[
\tan(5^\circ) = \frac{161 \text{ feet}}{BC}
\]\)
6. Solving for \(\(BC\)\), the distance from the boat to the lighthouse:
\(\[
BC = \frac{161 \text{ feet}}{\tan(5^\circ)}
\]\)
7. Substituting the Values:
- First, convert the angle to radians to find \(\(\tan(5^\circ)\)\), if we were calculating this manually.
- But based on our previous result, we have the value directly.
8. Calculating the Result:
\(\[
BC = 1840.2384207445762 \text{ feet}
\]\)
9. Final Answer:
- Therefore, the distance from the fishing boat to the base of the lighthouse is approximately:
\(\[
\boxed{1840.24 \text{ feet}}
\]\)
This completes the detailed solution for finding the distance from the boat to the base of the lighthouse given the height and the angle of elevation.
