Let's solve this problem step by step using trigonometry.
1. Understanding the Problem:
- We have a lighthouse and a sailboat.
- The angle of depression from the lighthouse operator to the sailboat is 16 degrees.
- The horizontal distance (distance along the base) from the base of the lighthouse to the sailboat is 120 feet.
2. Visual Representation:
- Imagine a right triangle where:
- The distance along the base to the sailboat is the adjacent side (120 feet).
- The distance from the lighthouse operator to the sailboat is the hypotenuse (which we need to find).
- The lighthouse and the line of sight to the sailboat form the right angle.
3. Using Trigonometry:
- In a right triangle, the cosine of an angle is defined as the adjacent side divided by the hypotenuse.
- Mathematically, this is:
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
- Here, [tex]\(\theta\)[/tex] is the angle of depression, which is 16 degrees.
- The adjacent side is 120 feet.
- Let [tex]\(d\)[/tex] be the distance from the lighthouse operator to the sailboat (the hypotenuse).
4. Set Up the Equation:
[tex]\[
\cos(16^\circ) = \frac{120}{d}
\][/tex]
5. Solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{120}{\cos(16^\circ)}
\][/tex]
6. Calculate:
- From trigonometric tables or a calculator, we know [tex]\(\cos(16^\circ) \approx 0.9613\)[/tex].
- Substitute this value:
[tex]\[
d = \frac{120}{0.9613} \approx 124.8
\][/tex]
7. Conclusion:
- Thus, the distance from the lighthouse operator to the sailboat is approximately 124.8 feet.
So, the lighthouse operator is about 124.8 feet from the sailboat.
