A boat is heading towards a lighthouse, where Madeline is watching from a vertical distance of 140 feet above the water. Madeline measures an angle of depression to the boat at point \( A \) to be \( 5^{\circ} \). At some later time, Madeline takes another measurement and finds the angle of depression to the boat (now at point \( B \) ) to be \( 50^{\circ} \). 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 the boat travels from point \( A \) to point \( B \), we'll analyze the situation using trigonometry. **Given:** - Height of Madeline above water, \( H = 140 \) feet. - Angle of depression to point \( A \), \( \theta_A = 5^\circ \). - Angle of depression to point \( B \), \( \theta_B = 50^\circ \). **Approach:** 1. **Understand the Angles of Depression:** - The angle of depression from Madeline to the boat is the same as the angle of elevation from the boat to Madeline due to alternate interior angles (assuming a horizontal line from Madeline). 2. **Set Up Right Triangles:** - For point \( A \): \[ \tan(\theta_A) = \frac{H}{x_A} \implies x_A = \frac{H}{\tan(\theta_A)} = \frac{140}{\tan(5^\circ)} \] - For point \( B \): \[ \tan(\theta_B) = \frac{H}{x_B} \implies x_B = \frac{H}{\tan(50^\circ)} = \frac{140}{\tan(50^\circ)} \] - The distance traveled by the boat, \( AB \), is the difference between these horizontal distances: \[ AB = x_A - x_B \] 3. **Calculate the Distances:** - Calculate \( \tan(5^\circ) \) and \( \tan(50^\circ) \) using a calculator: \[ \tan(5^\circ) \approx 0.0874887 \quad \text{and} \quad \tan(50^\circ) \approx 1.1917536 \] - Compute \( x_A \) and \( x_B \): \[ x_A \approx \frac{140}{0.0874887} \approx 1600.2 \text{ feet} \] \[ x_B \approx \frac{140}{1.1917536} \approx 117.7 \text{ feet} \] - Calculate \( AB \): \[ AB \approx 1600.2 \text{ feet} - 117.7 \text{ feet} = 1482.5 \text{ feet} \] **Final Answer:** The boat travels approximately **1,482.5 feet** from point \( A \) to point \( B \). **Answer:** After rounding, the distance from A to B is 1 482.5 feet.