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? Draw the triangle

To solve this problem, we can use trigonometry. We'll start by identifying the relevant elements and drawing the triangle.

### Step 1: Identify the elements of the triangle
1. **Height of the lighthouse (opposite side)**: 90 ft.
2. **Angle of depression**: 3°45'. The angle of depression is the angle formed between the horizontal line from the observer's eye level and the line of sight to the boat. This angle is the same as the angle of elevation from the boat to the top of the lighthouse due to alternate interior angles.

### Step 2: Convert the angle of depression to decimal form
The angle is given as 3°45'. We convert it to decimal form as follows:
- \( 45' = \frac{45}{60} = 0.75^\circ \)
- So, \( 3^\circ 45' = 3.75^\circ \)

### Step 3: Set up the right triangle
1. **Opposite side (height of the lighthouse)**: 90 ft.
2. **Adjacent side (distance from the lighthouse to the boat)**: We'll denote this as \( x \).
3. **Angle of elevation**: 3.75°.

### Step 4: Use the tangent function
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substitute the values:
\[
\tan(3.75^\circ) = \frac{90}{x}
\]
Solving for \( x \):
\[
x = \frac{90}{\tan(3.75^\circ)}
\]

### Step 5: Calculate the value of \( x \)
Let's calculate \( x \) using a calculator.

\[
x = \frac{90}{\tan(3.75^\circ)} \approx \frac{90}{0.0655} \approx 1373.28 \text{ ft}
\]

### Result
The distance from the lighthouse to the boat is approximately **1373.28 ft**.

### Drawing the triangle
Below is the description of the triangle you would draw:

1. Draw a horizontal line representing the sea level.
2. Draw a vertical line from the sea level up to represent the lighthouse height of 90 ft.
3. Draw a line from the top of the lighthouse to the horizontal line, forming the angle of 3.75° with the horizontal.
4. Label the height as 90 ft and the distance along the sea level as \( x \approx 1373.28 \) ft.
  
Would you like to see this drawn out or have any further questions? Here are some related problems to consider:

1. What would be the distance to the boat if the angle of depression were doubled?
2. How would the distance change if the lighthouse were 120 ft tall?
3. What if the angle of depression were 5° instead of 3°45'?
4. How would the calculation differ if the observer was looking at a plane in the sky instead of a boat?
5. Can we find the horizontal distance to the boat if we only know the lighthouse's height?

**Tip:** Remember that small changes in angle can lead to large changes in distance when dealing with tall structures and distant objects.

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? Draw the triangle.

Solve the trigonometry problem involving a 90 ft tall lighthouse and an angle of depression of 3°45'. Learn how to use the tangent function to find the distance to the boat and understand the key concepts behind angles of depression and right triangles. Suitable for high school students in Grades 9-12.

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