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 \( AB \) between points \( A \) and \( B \), we'll use trigonometric relationships based on the angles of elevation and the height of the beacon light. ### Given: - Height of the beacon light, \( h = 143 \) feet. - Angle of elevation from point \( A \), \( \theta_A = 5^\circ \). - Angle of elevation from point \( B \), \( \theta_B = 10^\circ \). ### Steps: 1. **Define Distances:** - Let \( x \) be the horizontal distance from point \( A \) to the base of the lighthouse. - Let \( d \) be the distance between points \( A \) and \( B \). - Therefore, the horizontal distance from point \( B \) to the base of the lighthouse is \( x - d \). 2. **Apply Tangent Function:** - From point \( A \): \[ \tan(5^\circ) = \frac{143}{x} \quad \Rightarrow \quad x = \frac{143}{\tan(5^\circ)} \] - From point \( B \): \[ \tan(10^\circ) = \frac{143}{x - d} \quad \Rightarrow \quad x - d = \frac{143}{\tan(10^\circ)} \] 3. **Express \( d \) in Terms of \( x \):** \[ d = x - \frac{143}{\tan(10^\circ)} = \frac{143}{\tan(5^\circ)} - \frac{143}{\tan(10^\circ)} \] 4. **Calculate Values:** - Calculate \( \tan(5^\circ) \) and \( \tan(10^\circ) \): \[ \tan(5^\circ) \approx 0.08749 \quad \text{and} \quad \tan(10^\circ) \approx 0.17633 \] - Substitute these values: \[ x \approx \frac{143}{0.08749} \approx 1634.3 \text{ feet} \] \[ x - d \approx \frac{143}{0.17633} \approx 811.0 \text{ feet} \] - Therefore: \[ d \approx 1634.3 - 811.0 = 823.3 \text{ feet} \] ### Conclusion: The distance from point \( A \) to point \( B \) is **approximately 823.3 feet**. **Answer:** Approximately 823.3 feet of distance separates points A and B.