A lighthouse is 90 ft above sea level. A person at the top of the lighthouse observes a boat in the water at an angle of depression of 3°45'. What is the distance from the lighthouse to the boat? And illustrate the triangle 

To solve this problem, we can use trigonometry. Specifically, we'll apply the tangent function because we know the angle of depression and the height of the lighthouse, and we need to find the horizontal distance (adjacent side) between the lighthouse and the boat.

### Given:
- Height of the lighthouse = 90 ft (opposite side of the triangle)
- Angle of depression = 3°45' (convert this to decimal form)

First, convert the angle from degrees and minutes to decimal form:
\[
3^\circ 45' = 3^\circ + \frac{45}{60} = 3.75^\circ
\]

Now, use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \(\theta = 3.75^\circ\)
- Opposite = 90 ft (height of the lighthouse)
- Adjacent = distance from the lighthouse to the boat (what we are solving for)

Rearranging for the adjacent side (distance to the boat):
\[
\text{Adjacent} = \frac{\text{opposite}}{\tan(\theta)} = \frac{90}{\tan(3.75^\circ)}
\]

Let me calculate this.The horizontal distance from the lighthouse to the boat is approximately 1,373.13 feet.

### To illustrate the triangle:
1. The lighthouse forms a vertical line, representing the **opposite** side of the triangle, which is 90 ft.
2. The **adjacent** side, which is the horizontal distance from the base of the lighthouse to the boat, is about 1,373.13 ft.
3. The **hypotenuse** is the line of sight from the top of the lighthouse to the boat, making an angle of depression of \(3^\circ 45'\) with the horizontal line.

Would you like me to generate a detailed illustration of the triangle?

---

Here are some related questions you might consider:
1. How does the angle of depression affect the distance to the boat?
2. What if the height of the lighthouse were doubled, how would the distance change?
3. How do you find the hypotenuse length of this right triangle?
4. How would the distance change if the angle of depression increased to \(5^\circ\)?
5. What trigonometric function would you use if the hypotenuse were known instead of the height?

**Tip:** Always convert angles in degrees and minutes to decimal form when using trigonometric functions!

A lighthouse is 90 ft above sea level. A person at the top of the lighthouse observes a boat in the water at an angle of depression of 3°45'. What is the distance from the lighthouse to the boat? And illustrate the triangle.

Learn how to calculate the distance from a lighthouse to a boat using trigonometric principles. This problem involves an angle of depression of 3°45', a lighthouse height of 90 ft, and the application of the tangent function. Suitable for students in Grades 9-11.

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