Final answer:
The piecewise function models the distance traveled at different speeds over time on a trip. Without the decay parameter, the probability of commuting more than 25 miles cannot be calculated. The concepts of uniform motion and changes in velocity are illustrated through examples and equations, such as position vs. time graphs and the relationship between displacement, average velocity, and time.
Step-by-step explanation:
The piecewise function described represents the total distance traveled over time, with two different rates based on the duration of travel. For the first segment (0≤x≤2 hours), the function is 55x, indicating that the speed was 55 miles per hour. For the second segment (2<x≤5 hours), the function is 65x-20, suggesting an increase in speed to 65 miles per hour, but with a 20-mile deduction, likely due to a stop or a slower rate earlier in the trip.
To calculate the probability that a person is willing to commute more than 25 miles if the distance is an exponential random variable with a decay parameter, you would typically use the formula of the exponential distribution. However, since the specific decay parameter is not provided in the question, the precise probability cannot be calculated here. The variables m, µ, and o in the context of exponential distributions typically refer to the scale, mean, and standard deviation, respectively.
Related to motion, the uniform motion and changes in velocity are exemplified in different scenarios. A uniform speed results in a linear relationship between distance and time, as shown by a constant slope on a position vs. time graph. When a vehicle stops or changes speed, the graph curves, being concave downward or upward depending on deceleration or acceleration.
In the example of a train starting from rest, accelerating, maintaining speed, and then decelerating, the corresponding position vs. time graph would show a concave upward curve during acceleration, a straight line during constant velocity, and a concave downward curve during deceleration. The linear portion of the graph would have a slope of 100 miles per hour, representing the constant speed.
Calculating average velocity and displacement over a time period is another concept addressed through the equation x = xo + ut, which shows that for a constant velocity, the displacement is a linear function of time. The longer the time or the higher the velocity, in straight proportion, the greater the distance covered.