A boat is heading towards a lighthouse, whose beacon-light is 143 feet above the water.
From point , the boat’s crew measures the angle of elevation to the beacon, , before
they draw closer. They measure the angle of elevation a second time from point to be
. Find the distance from point to point . Round your answer to the nearest tenth
of a foot if necessary.

Ask by Floyd Gardner. in the United States

Jan 27,2025

Question

A boat is heading towards a lighthouse, whose beacon-light is 143 feet above the water.
From point , the boat’s crew measures the angle of elevation to the beacon, , before
they draw closer. They measure the angle of elevation a second time from point to be
. Find the distance from point to point . Round your answer to the nearest tenth
of a foot if necessary.

Ask by Floyd Gardner. in the United States

Jan 27,2025

UpStudy AI · Accepted Answer

To determine the distance $$ AB $$ that the boat traveled from point $$ A $$ to point $$ B $$, we'll use trigonometric relationships involving the angles of elevation from each point to the lighthouse's beacon.

### Given:
- **Height of beacon (h):** 143 feet
- **Angle of elevation from point $$ A $$:** $$ 5^\circ $$
- **Angle of elevation from point $$ B $$:** $$ 10^\circ $$

### Steps:

1. **Determine Horizontal Distances from Lighthouse:**
   
   Let's denote:
   - $$ AL $$ as the horizontal distance from point $$ A $$ to the lighthouse.
   - $$ BL $$ as the horizontal distance from point $$ B $$ to the lighthouse.

Using the tangent of the angle of elevation:
   $$
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\text{horizontal distance}}
   $$
   
   Therefore:
   $$
   AL = \frac{h}{\tan(5^\circ)}
   $$
   $$
   BL = \frac{h}{\tan(10^\circ)}
   $$

2. **Calculate $$ AL $$ and $$ BL $$:**
   
   - **For $$ AL $$:**
     $$
     \tan(5^\circ) \approx 0.08749
     $$
     $$
     AL = \frac{143}{0.08749} \approx 1634.3 \text{ feet}
     $$
   
   - **For $$ BL $$:**
     $$
     \tan(10^\circ) \approx 0.17633
     $$
     $$
     BL = \frac{143}{0.17633} \approx 811.2 \text{ feet}
     $$

3. **Determine Distance $$ AB $$:**
   
   Since the boat is moving directly towards the lighthouse:
   $$
   AB = AL - BL
   $$
   $$
   AB = 1634.3 - 811.2 = 823.1 \text{ feet}
   $$

4. **Final Answer:**

Rounded to the nearest tenth of a foot, the distance $$ AB $$ is:

$$
   AB \approx 823.0 \text{ feet}
   $$

### Conclusion:

The distance from point $$ A $$ to point $$ B $$ is approximately **823.0 feet**.

**Answer:**  
The distance from point A to point B is approximately 823.0 feet.
The distance from point A to point B is approximately 823.0 feet.

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