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 [tex]\( A \)[/tex].
- Label the bottom of the lighthouse (sea level) as point [tex]\( B \)[/tex].
- Label the position of the boat as point [tex]\( C \)[/tex].
[tex]\[
\text{A}(Lighthouse)
\][/tex]
[tex]\[
|
\][/tex]
[tex]\[
|
\][/tex]
[tex]\[
|
\][/tex]
[tex]\[
| (Height = 161 \text{ feet})
\][/tex]
[tex]\[
|
\][/tex]
[tex]\[
|
\][/tex]
[tex]\[
B = \text{Base of lighthouse}
\][/tex]
[tex]\[
\_\_\_\_\_\_\_\_\_\_\_\_ \][/tex] [tex]\(\angle = 5^\circ \text{ (Angle of elevation)}\)[/tex]
[tex]\[
C \text{ (Boat)}
\][/tex]
3. Setting up the problem:
- The height from the base to the top of the lighthouse ([tex]\(AB\)[/tex]) is 161 feet (opposite side).
- The distance from the boat to the base of the lighthouse ([tex]\(BC\)[/tex]) is the adjacent side.
- The angle of elevation ([tex]\(\angle BAC\)[/tex]) 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:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, [tex]\(\theta\)[/tex] is 5°, the opposite side is 161 feet.
5. Formulating the Equation:
[tex]\[
\tan(5^\circ) = \frac{161 \text{ feet}}{BC}
\][/tex]
6. Solving for [tex]\(BC\)[/tex], the distance from the boat to the lighthouse:
[tex]\[
BC = \frac{161 \text{ feet}}{\tan(5^\circ)}
\][/tex]
7. Substituting the Values:
- First, convert the angle to radians to find [tex]\(\tan(5^\circ)\)[/tex], if we were calculating this manually.
- But based on our previous result, we have the value directly.
8. Calculating the Result:
[tex]\[
BC = 1840.2384207445762 \text{ feet}
\][/tex]
9. Final Answer:
- Therefore, the distance from the fishing boat to the base of the lighthouse is approximately:
[tex]\[
\boxed{1840.24 \text{ feet}}
\][/tex]
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.
