Airline travel The following figure shows the ​​position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.

Verified step by step guidance

1. To determine the velocity of the airliner at noon (t = 6), we need to find the derivative of the position function s = f(t) with respect to time t. The derivative, f'(t), represents the velocity of the airliner at any given time t.
2. Evaluate the derivative f'(t) at t = 6 to find the velocity at noon. This involves substituting t = 6 into the derivative function f'(t).
3. The velocity is negative because the airliner is on its return trip to Seattle. A negative velocity indicates that the direction of travel is towards the starting point, which in this case is Seattle.
4. Conceptually, velocity is a vector quantity, meaning it has both magnitude and direction. A negative value signifies movement in the opposite direction to the positive axis, which is typically defined as the direction away from the starting point.
5. Ensure that the units of the velocity are consistent with the units used in the position function. If the position function is in miles, the velocity will be in miles per hour.

Key Concepts

Position Function

The position function, denoted as s = f(t), describes the location of an object over time. In this context, it represents the distance of the airliner from Seattle as a function of time since take-off. Understanding this function is crucial for analyzing the motion of the airliner, as it provides the necessary data to determine both velocity and acceleration.

Velocity

Velocity is defined as the rate of change of position with respect to time, mathematically expressed as v(t) = f'(t). It indicates both the speed and direction of an object's movement. In this scenario, calculating the velocity at a specific time, such as noon (t = 6), helps to understand how fast the airliner is traveling and in which direction relative to Seattle.

Negative Velocity

A negative velocity indicates that the object is moving in the opposite direction to the defined positive direction, which in this case is towards Seattle. When the airliner is returning to Seattle after reaching Minneapolis, its velocity becomes negative. This concept is essential for interpreting the motion of the airliner and understanding the implications of its position function at different times.