A hot-air balloon is flying at an altitude of 2589 feet. If the angle of depression from the pilot in the balloon to a house on the ground below is \( 32^{\circ} \), how far is the house from the pilot? The house is feet from the pilot of the balloon. (Do not round until the final answer. Then round to the nearest tenth as needed.)

To determine the distance from the house to the pilot, we can use trigonometry. The angle of depression is the angle between the horizontal line from the balloon and the line of sight to the house. First, we can identify the opposite side (the altitude of the balloon) and the adjacent side (the distance to the house) for the right triangle formed. Using the tangent function, which relates these sides as: \[ \tan(32^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2589}{d} \] where \(d\) is the distance from the balloon to the house. Re-arranging the formula gives: \[ d = \frac{2589}{\tan(32^\circ)} \] Calculating this, \[ d \approx \frac{2589}{0.62487} \approx 4142.33 \text{ feet} \] Thus, rounded to the nearest tenth, the house is approximately **4142.3 feet** from the pilot of the balloon. In a hot-air balloon, the pilot has a unique perspective. They can float high above the ground and observe everything from a bird's-eye view, which can be thrilling! Did you know that hot-air ballooning has been around since the late 18th century? The first passengers in a balloon were a sheep, a duck, and a rooster, launched by the Montgolfier brothers in France. This math problem can be tackled by understanding basic trigonometric principles. Be cautious with your calculator settings; ensure it's on the correct mode (degrees, not radians) when using functions like tangent! One common mistake is mixing up the opposite and adjacent sides. Always visualize or sketch a right triangle to avoid confusion.