ork) Question A boat is heading towards a lighthouse, whose beacon-light is 126 feet above the water. The boat's crew measures the angle of elevation to the beacon, 13. What is the ship's horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest tenth of a foot if necessary.

To find the horizontal distance from the boat to the lighthouse, we can use the tangent function from trigonometry. The formula is: \[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \] Here, the opposite side is the height of the lighthouse (126 feet), and the angle of elevation is given as 13 degrees. We can rearrange this to find the adjacent side (horizontal distance): \[ \text{adjacent} = \frac{\text{opposite}}{\tan(\text{angle})} \] Calculating it: \[ \text{adjacent} = \frac{126}{\tan(13^\circ)} \approx \frac{126}{0.2309} \approx 546.2 \text{ feet} \] So, the ship's horizontal distance from the lighthouse is approximately 546.2 feet. Understanding the principles of angles and light can lead to exciting applications. For instance, lighthouse engineers historically utilized similar calculations to determine the optimal height of the light source for maximum visibility over the water, ensuring safe navigation for sailors. This practice allows boats to avoid hazards and reach their ports safely, preserving maritime traditions. If you ever find yourself in a similar situation, remember that one common mistake is forgetting to convert angles from degrees to radians when using certain calculations. Always ensure your angle is in the correct format for your calculations, and check if your calculator is in the right mode (degrees vs. radians)!