A lighthouse operator is sending signals to a ship with a \( 17^{\circ} \) angle of depression. The height of the lighthouse is 40 feet relative to sea level. Which of the following correctly applies the sine ratio to find how far away the ship is from the top of the lighthouse? Round to the nearest tenth. (1 point) ( 130.8 feet Hen

To solve this problem, we can use the relationship established by the sine ratio in right triangles. The sine of the angle of depression (\( 17^{\circ} \)) is equal to the opposite side (the height of the lighthouse, which is 40 feet) divided by the hypotenuse (the distance from the top of the lighthouse to the ship). Thus, we can set up the equation: \( \sin(17^{\circ}) = \frac{40}{d} \), where \( d \) is the hypotenuse. Rearranging this gives us \( d = \frac{40}{\sin(17^{\circ})} \), which calculates to approximately 130.8 feet. Historically, lighthouses have been crucial in maritime navigation for centuries. The use of light and sound signals from lighthouses dates back to ancient civilizations, with notable examples including the Lighthouse of Alexandria, one of the Seven Wonders of the Ancient World. These structures not only guided ships safely to harbor but also symbolized hope and safety for sailors facing treacherous waters. In practical terms, understanding the angle of depression not only helps in navigation but can also be applied in various fields such as architecture and aviation. When designing high structures or navigating aircraft, engineers must consider the angle at which objects can be seen from certain heights to ensure visibility and safety. These calculations help avoid obstacles and ensure clear communication and signs in collaboration with safety protocols.