An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If $t$ represents the time in minutes since the plane has left the terminal building, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane.(a) Sketch a possible graph of $x(t).$(b) Sketch a possible graph of $y(t).$(c) Sketch a possible graph of the ground speed.(d) Sketch a possible graph of the vertical velocity.

Key Concepts: Graphs, Distance, Velocity, Altitude Explanation: This problem describes an airplane that takes off from an airport and lands an hour later at another airport, 400 miles away. We are asked to sketch possible graphs of x(t), y(t), ground speed, and vertical velocity based on the given information. Step by Step Solution: Step 1. First, let's consider the graph of x(t). The airplane travels a horizontal distance of 400 miles in 60 minutes, so its average speed is 400/60 = 20/3 miles per minute. This means that the graph of x(t) should be a line with a slope of 20/3, since distance is equal to rate times time. The line should start at x(0) = 0 and end at x(60) = 400. Step 2. Next, let's consider the graph of y(t). We are not given any information about the altitude of the plane, so the graph of y(t) could take on many different shapes depending on the flight path of the airplane. However, we know that the plane takes off and lands, so the graph should start and end at y equals some value, and it should have some peaks and valleys. Step 3. Now let's consider the ground speed. Since the airplane travels at a constant speed of 20/3 miles per minute horizontally, the ground speed will be equal to this value for the entire trip. This means the graph of ground speed should be a flat line at 20/3 miles per minute. Step 4. Finally, let's consider the vertical velocity. We are not given any information about the direction or speed of the airplane's ascent or descent, so the graph of vertical velocity could take on many different shapes depending on the flight path of the airplane. However, we know that the plane takes off and lands, so the graph should start and end at a velocity of zero.