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A boat is heading towards a lighthouse, whose beacon-light is 145 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 11degrees
∘
. What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest tenth of a foot if necessary.
Answer
Attempt 1 out of 2

To find the horizontal distance from the boat to the lighthouse, we can use trigonometry. Specifically, we'll use the tangent function, which relates the angle of elevation to the opposite side (the height of the lighthouse) and the adjacent side (the horizontal distance from the boat to the lighthouse).

### Step 1: Identify the known values.
- Height of the lighthouse (opposite side): \( h = 145 \) feet
- Angle of elevation: \( \theta = 11^\circ \)

### Step 2: Write the equation using the tangent function.
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:

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

Where:
- \( d \) is the horizontal distance from the boat to the lighthouse (the adjacent side).

### Step 3: Solve for the horizontal distance \( d \).
\[
d = \frac{h}{\tan(\theta)}
\]

Substituting the known values:

\[
d = \frac{145}{\tan(11^\circ)}
\]

### Step 4: Calculate the value.
Using a calculator:

\[
d \approx \frac{145}{\tan(11^\circ)} \approx \frac{145}{0.1944} \approx 746.0 \text{ feet}
\]

### Final Answer:
The ship's horizontal distance from the lighthouse is approximately **746.0 feet**.

Would you like more details or have any questions?

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Here are some related questions to consider:
1. How would the distance change if the angle of elevation were increased to 15 degrees?
2. What would be the distance if the lighthouse were twice as tall?
3. How does the calculation change if the boat is moving and the angle of elevation is increasing?
4. How would you calculate the distance if the height of the lighthouse was not given but the angle and distance were?
5. What if the angle of elevation was measured as 11.5 degrees instead of 11 degrees?
6. Can you determine the distance using the sine or cosine functions instead of tangent?
7. What is the height of the lighthouse if the angle is 11 degrees and the distance is 800 feet?
8. How would this problem change if the lighthouse were on a hill above the water?

**Tip:** Always check that your calculator is set to the correct mode (degrees or radians) when working with trigonometric functions.

Learn how to calculate the horizontal distance from a boat to a lighthouse using trigonometry. Given the lighthouse's height of 145 feet and an angle of elevation of 11 degrees, find the boat's horizontal distance to the lighthouse shore. Follow a step-by-step solution involving the tangent function. Suitable for Grades 9-12.

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