Create a piecewise function of the following scenario where D(t) is the total distance walked (in miles) as a function of time t (in minutes):
You begin a walk from your home at a fast pace of 6 miles per hour for 30 minutes. You rest for 10 minutes, and then you continue walking at a leisurely pace of 1 mile per hour for 25 minutes."
D(t)= __?__ if 30 < t </ 40.
D(t)= __?__ if 40 < t </65.
6*1/2 = 3 while you rest
3 + (1/60 mile/min)(t-40)
To create a piecewise function for this scenario, we need to break it down into different sections based on the given time intervals.
Let's break it down step-by-step:
1. The first segment is from t = 0 to t = 30 minutes, during which you walk at a fast pace of 6 miles per hour.
To calculate the distance for this segment, we can use the formula: distance = speed * time. Since the speed is given in miles per hour and the time is in minutes, we need to convert the speed to miles per minute.
Since there are 60 minutes in an hour, the speed in miles per minute will be (6 miles per hour) / (60 minutes per hour) = 0.1 miles per minute.
So, for the first segment, which covers t values from 0 to 30 minutes, the distance walked is D(t) = (0.1 miles per minute) * t.
2. The second segment is from t = 30 to t = 40 minutes, during which you rest and do not walk. The distance covered during this time will be 0 miles.
So, for the second segment, which covers t values from 30 to 40 minutes, the distance walked is D(t) = 0.
3. The third segment is from t = 40 to t = 65 minutes, during which you walk at a leisurely pace of 1 mile per hour.
Similarly, we convert the speed to miles per minute: (1 mile per hour) / (60 minutes per hour) = 0.0167 miles per minute.
So, for the third segment, which covers t values from 40 to 65 minutes, the distance walked is D(t) = (0.0167 miles per minute) * (t - 40 minutes).
Putting it all together, the piecewise function for D(t) is:
D(t) =
0.1t, if 0 < t ≤ 30
0, if 30 < t ≤ 40
0.0167(t - 40), if 40 < t ≤ 65
Note: We consider the intervals with "<=" for inclusive bounds and "<" for exclusive bounds.