A boat is heading towards a lighthouse, whose beacon-light is 126 feet above the water. From point $A$, the boat's crew measures the angle of elevation to the beacon, $12^{\circ}$, before they draw closer. They measure the angle of elevatiol a second time from point $B$ at some later time to be $25^{\circ}$. Find the distance from point $A$ to point $B$. Round your answer to the nearest foot if necessary
Answer
\(\boxed{323}\) feet is the distance between point A and point B.
Step 1 ：Let the distance from point A to the lighthouse be $x$ and the distance from point B to the lighthouse be $y$. We have the following equations:
Step 2 ：\(\tan(12^\circ) = \frac{126}{x}\)
Step 3 ：\(\tan(25^\circ) = \frac{126}{y}\)
Step 4 ：Solve the equations for $x$ and $y$:
Step 5 ：\(x = \frac{126}{\tan(12^\circ)} \approx 592.78\)
Step 6 ：\(y = \frac{126}{\tan(25^\circ)} \approx 270.21\)
Step 7 ：Find the distance between points A and B, which is $x - y$:
Step 8 ：\(x - y \approx 592.78 - 270.21 \approx 322.57\)
Step 9 ：\(\boxed{323}\) feet is the distance between point A and point B.
