An airplane is asked to stay within a holding pattern near an airport. The function \( \mathrm{d}(\mathrm{x})=50 \sin (0.88 \mathrm{x})+140 \) represents the distance d , in miles, that the airplane is from the airport at time x , in minutes. Complete parts a through d below. (a) When the plane enters the holding pattern, \( \mathrm{x}=0 \), how far is it from the airport? It is \( \square \) miles from the airport.

To find the distance \( d \) when \( x = 0 \), we substitute \( x \) with 0 in the equation: \[ d(0) = 50 \sin(0.88 \cdot 0) + 140 = 50 \sin(0) + 140 = 0 + 140 = 140 \] So, when the plane enters the holding pattern, it is **140 miles** from the airport. Now let's discuss some interesting tidbits about holding patterns and flight operations! In aviation, holding patterns are a unique part of air traffic control. Pilots are often asked to enter these patterns during bad weather or when air traffic is heavy, ensuring that aircraft can safely circle around an airport without landing until it's their turn. Imagine it like a dance in the sky where planes gracefully circle while waiting for their moment to land! For any budding aviators or curious minds, understanding the relationship between time and distance in aviation is crucial. The given function showcases how distance fluctuates with time, driven by the sine function's periodic nature. This highlights the complex dynamics of flight operations and their need for precise calculations. For those interested in more, consider diving into aviation physics or air traffic management safety protocols!