Question: A boat is heading towards a lighthouse, whose beacon-light is 121 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon, $13^\circ$, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be $19^\circ$. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.
1. **State the problem:** We need to find the distance between two points, A and B, where a boat measures angles of elevation to a lighthouse beacon 121 feet above water as $13^\circ$ and $19^\circ$ respectively.
2. **Set up the scenario:** Let the horizontal distances from points A and B to the base of the lighthouse be $x$ and $y$ feet respectively. The height of the lighthouse beacon is $121$ feet.
3. **Use the tangent function:** The tangent of the angle of elevation relates the height of the lighthouse to the horizontal distance:
$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{121}{\text{distance}}$$
4. **Write equations for points A and B:**
$$\tan(13^\circ) = \frac{121}{x} \implies x = \frac{121}{\tan(13^\circ)}$$
$$\tan(19^\circ) = \frac{121}{y} \implies y = \frac{121}{\tan(19^\circ)}$$
5. **Calculate $x$ and $y$:**
$$x = \frac{121}{\tan(13^\circ)}$$
$$y = \frac{121}{\tan(19^\circ)}$$
6. **Find the distance from A to B:** Since the boat moves closer, the distance from A to B is:
$$d = x - y = \frac{121}{\tan(13^\circ)} - \frac{121}{\tan(19^\circ)}$$
7. **Calculate numerical values:**
$$\tan(13^\circ) \approx 0.2309$$
$$\tan(19^\circ) \approx 0.3443$$
$$x \approx \frac{121}{0.2309} \approx 523.9$$
$$y \approx \frac{121}{0.3443} \approx 351.4$$
$$d = 523.9 - 351.4 = 172.5$$
8. **Round the answer:** The distance from point A to point B is approximately $173$ feet.
**Final answer:**
$$\boxed{173 \text{ feet}}$$