From the observation deck of the lighthouse at Sasquatch Point 30 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the lighthouse. The first sighting had an angle of depression of 8.5° and the second sighting had an angle of depression of 25.7°. How far had the boat traveled between the sightings? Round your answer to the nearest foot.


To find how far the boat traveled between the sightings, we need to apply trigonometric concepts, specifically using the tangent of the angles of depression.

### Step-by-Step Solution:

1. **Define Variables:**
   - Height of the observation deck above the water: \( h = 30 \) feet.
   - The first angle of depression: \( \theta_1 = 8.5^\circ \).
   - The second angle of depression: \( \theta_2 = 25.7^\circ \).

2. **Set Up Equations:**
   Since the angles of depression are measured from the horizontal, we can use the tangent function to relate the angles of depression, the height of the observation deck, and the horizontal distances.

   Let:
   - \( d_1 \) be the horizontal distance from the lighthouse to the boat at the first sighting.
   - \( d_2 \) be the horizontal distance from the lighthouse to the boat at the second sighting.
   
   From the tangent definition, we have:
   \[
   \tan(\theta_1) = \frac{h}{d_1} \quad \text{and} \quad \tan(\theta_2) = \frac{h}{d_2}
   \]

   Substituting values:
   \[
   \tan(8.5^\circ) = \frac{30}{d_1} \quad \text{and} \quad \tan(25.7^\circ) = \frac{30}{d_2}
   \]

3. **Solve for \( d_1 \) and \( d_2 \):**

   Using a calculator:
   \[
   d_1 = \frac{30}{\tan(8.5^\circ)}
   \quad \text{and} \quad
   d_2 = \frac{30}{\tan(25.7^\circ)}
   \]

Let me calculate these values.The boat traveled approximately 138 feet between the two sightings.

Would you like more details or have any questions about this process? Here are five related questions you might explore:

1. How would the calculation change if the observation deck were higher?
2. What if the boat was moving at a constant speed; how long would it take to travel this distance?
3. How would you calculate the boat's speed if the times of the sightings were given?
4. Can this method be used for distances greater than 500 feet?
5. How would changing the angles of depression affect the outcome?

**Tip:** Always double-check whether your angles are in degrees or radians when using trigonometric functions in calculations!

From the observation deck of the lighthouse at Sasquatch Point 30 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the lighthouse. The first sighting had an angle of depression of 8.5° and the second sighting had an angle of depression of 25.7°. How far had the boat traveled between the sightings? Round your answer to the nearest foot.

Solve this trigonometry problem involving angles of depression from a lighthouse to calculate how far a boat traveled on Lake Ippizuti. Learn to apply the tangent function to real-life situations with step-by-step guidance. Suitable for high school students, Grades 9-12.

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