A boat is heading towards a lighthouse, whose beacon-light is 143 feet above the water. From point \( A \), the boat's crew measures the angle of elevation to the beacon, \( 5^{\circ} \), before they draw closer. They measure the angle of elevation a second time from point \( B \) to be \( 10^{\circ} \). Find the distance from point \( A \) to point \( B \). Round your answer to the nearest tenth of a foot if necessary.

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.