Question: A boat is heading towards a lighthouse, where Meena is watching from a vertical distance of $120$ feet above the water. Meena measures an angle of depression to the boat at point A to be $18^\circ$. At some later time, Meena takes another measurement and finds the angle of depression to the boat (now at point B) to be $44^\circ$. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.
1. **Problem Statement:**
Meena is observing a boat from a lighthouse $120$ feet above water. The angles of depression to the boat at two points, A and B, are $18^\circ$ and $44^\circ$ respectively. We need to find the distance between points A and B along the water.
2. **Diagram and Setup:**
Let the horizontal distances from the lighthouse base to points A and B be $x$ and $y$ respectively, with $x > y$ since the boat is moving closer.
3. **Using Trigonometry:**
The angle of depression equals the angle between the horizontal line from Meena's eye level and the line of sight to the boat.
From the right triangles formed:
$$\tan(18^\circ) = \frac{120}{x} \implies x = \frac{120}{\tan(18^\circ)}$$
$$\tan(44^\circ) = \frac{120}{y} \implies y = \frac{120}{\tan(44^\circ)}$$
4. **Calculate $x$ and $y$:**
Calculate $\tan(18^\circ)$ and $\tan(44^\circ)$:
$$\tan(18^\circ) \approx 0.3249$$
$$\tan(44^\circ) \approx 0.9657$$
So,
$$x = \frac{120}{0.3249} \approx 369.3$$
$$y = \frac{120}{0.9657} \approx 124.3$$
5. **Find the distance between points A and B:**
$$\text{Distance} = x - y = 369.3 - 124.3 = 245$$
6. **Final Answer:**
The distance from point A to point B is approximately **245 feet**.