Question

A lighthouse keeper at 180 feet off the ground can see a boat at a 25° angle of depression. How far from the base of the lighthouse is the boat?

Answer 1

To find the distance from the base of the lighthouse to the boat, we can use trigonometry. In this case, we can use the tangent of the angle of depression.

1. **Identify the variables**:
- Height of the lighthouse (\(h\)) = 180 feet
- Angle of depression (\(\theta\)) = 25°

2. **Understand the scenario**:
The angle of depression from the lighthouse keeper to the boat is the same as the angle of elevation from the boat to the lighthouse keeper, both being 25°. Therefore, we can use the tangent function:

\[
\tan(\theta) = \frac{{\text{{opposite}}}}{{\text{{adjacent}}}} = \frac{h}{d}
\]

where \(d\) is the horizontal distance from the base of the lighthouse to the boat.

3. **Set up the equation**:
\[
\tan(25^\circ) = \frac{180}{d}
\]

4. **Rearrange to solve for \(d\)**:
\[
d = \frac{180}{\tan(25^\circ)}
\]

5. **Calculate**:
- First, find \(\tan(25^\circ)\) using a calculator:
\[
\tan(25^\circ) \approx 0.4663
\]

- Now substitute this value into the equation:
\[
d \approx \frac{180}{0.4663} \approx 385.4 \text{ feet}
\]

Therefore, the boat is approximately **385.4 feet** from the base of the lighthouse.