From the top of a lighthouse 70 feet high, the angle of depression to a smaller boat is 38.9 degrees.  How far from the foot of the lighhouse is the boat?

We can solve this problem using trigonometry, specifically by applying the tangent function. The angle of depression and the height of the lighthouse form a right triangle with the distance from the foot of the lighthouse to the boat.

### Given:
- Height of the lighthouse = 70 feet
- Angle of depression = 38.9 degrees
- We are asked to find the horizontal distance from the foot of the lighthouse to the boat (let's call this distance \( x \)).

### Tangent function:
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the lighthouse (70 feet) and the adjacent side is the distance from the lighthouse to the boat ( \( x \)).

\[
\tan(38.9^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{70}{x}
\]

### Rearranging to solve for \( x \):

\[
x = \frac{70}{\tan(38.9^\circ)}
\]

Now let's calculate \( x \). 

The distance from the foot of the lighthouse to the boat is approximately 86.75 feet.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the distance change if the angle of depression increased?
2. What if the height of the lighthouse was 100 feet instead of 70 feet?
3. How can we calculate the distance if the boat moves closer, reducing the angle of depression to 25 degrees?
4. How can we find the height of the lighthouse if the distance to the boat is known?
5. What other trigonometric functions can be used to solve similar problems?

**Tip:** The angle of depression is always measured from the horizontal line, not from the vertical.

From the top of a lighthouse 70 feet high, the angle of depression to a smaller boat is 38.9 degrees. How far from the foot of the lighthouse is the boat?

Learn how to calculate the horizontal distance from a 70-foot tall lighthouse to a boat using the angle of depression of 38.9 degrees. This problem applies trigonometric concepts like the tangent function to find the solution. Suitable for Grades 9-10.

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