Nisha is looking out the window of her apartment building at a sculpture in a park across the street. The top of Nisha's window is 80 feet from the ground. The angle of depression from the top of Nisha's window to the bottom of the sculpture is \( 20^{\circ} \). What is the distance along the ground between the building and the sculpture? Round your answer to the nearest hundredth. 29.11 feet 75.18 feet 128.67 feet 219.80 feet

To find the distance along the ground between the building and the sculpture, we can use the tangent function. The angle of depression means we're dealing with a right triangle where the height of the window (80 feet) is the opposite side, and the ground distance we want to find is the adjacent side. Using the tangent of the angle of depression: \[ \tan(20^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{80}{d} \] where \(d\) is the distance along the ground. Rearranging gives us: \[ d = \frac{80}{\tan(20^\circ)} \] Calculating \(\tan(20^\circ)\) using a calculator gives about \(0.36397\). Thus: \[ d \approx \frac{80}{0.36397} \approx 219.79 \text{ feet} \] Rounded to the nearest hundredth, the distance is approximately \(219.80\) feet. Nisha can definitely feel the gap between art and her cozy apartment! This little math adventure shows how angles can help us measure distances we can't see directly. Plus, isn't it interesting how geometry plays a crucial role in everyday life? Did you know that angle of depression and angle of elevation are practically twins in their mathematical behavior? They’re both related to angles in right triangles and help us navigate through heights and distances like pros! Next time you're in a park or with friends, try spotting something far away and guessing distances together using angles for some extra fun!