Question: A boat is heading towards a lighthouse, where Jeremiah is watching from a vertical distance of $126$ feet above the water. Jeremiah measures an angle of depression to the boat at point A to be $11^\circ$. At some later time, Jeremiah takes another measurement and finds the angle of depression to the boat (now at point B) to be $78^\circ$. Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.
1. **Problem Statement:**
Jeremiah is $126$ feet above water in a lighthouse. He measures angles of depression to a boat at two points, A and B, as $11^\circ$ and $78^\circ$ respectively. We need to find the distance between points A and B along the water.
2. **Setup and Formula:**
Let the horizontal distances from the lighthouse base to points A and B be $x$ and $y$ respectively.
Using the angle of depression and the height, we apply the tangent function:
$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{126}{\text{distance}}$$
So,
$$x = \frac{126}{\tan(11^\circ)}$$
$$y = \frac{126}{\tan(78^\circ)}$$
3. **Calculate $x$ and $y$:**
$$x = \frac{126}{\tan(11^\circ)}$$
Calculate $\tan(11^\circ)$:
$$\tan(11^\circ) \approx 0.1944$$
So,
$$x = \frac{126}{0.1944} \approx 647.9 \text{ feet}$$
Similarly for $y$:
$$\tan(78^\circ) \approx 4.7046$$
$$y = \frac{126}{4.7046} \approx 26.8 \text{ feet}$$
4. **Find distance between points A and B:**
The boat moves closer, so distance from A to B is:
$$d = x - y = 647.9 - 26.8 = 621.1 \text{ feet}$$
5. **Final answer:**
The distance from point A to point B is approximately **621.1 feet**.