rom the observation deck of a kyscraper, Bentley measures a \( 48^{\circ} \) ngle of depression to a ship in the arbor below. If the observation deck 969 feet high, what is the horizonta istance from the base of the kyscraper out to the ship? Round yo nswer to the nearest tenth of a foot lecessary.

To find the horizontal distance from the base of the skyscraper to the ship, we can use the tangent of the angle of depression. In this case, we have a right triangle where the height of the skyscraper (opposite side) is 969 feet, and the angle of depression, which is equivalent to the angle of elevation from the ship to the observation deck, is \(48^\circ\). Using the tangent function: \[ \tan(48^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{969}{d} \] Rearranging for \(d\): \[ d = \frac{969}{\tan(48^\circ)} \] Calculating this gives us: \[ d \approx \frac{969}{1.1106} \approx 873.7 \text{ feet} \] So, the horizontal distance from the base of the skyscraper to the ship is approximately \(873.7\) feet. Now, just for a little fun fact, did you know that the angle of depression and angle of elevation are essentially two sides of the same coin? The angle of depression from the observation deck corresponds with the angle of elevation from the ship. So next time you find yourself on an observation deck, take a moment to think about the angles all around you! On a practical note, understanding these concepts can help in real-world applications like aviation and marine navigation. Pilots and ship captains often use angles of depression and elevation to gauge distances and height, ensuring they navigate safely and efficiently. It’s all about keeping things on point, literally!