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 25 minutes. You rest for 15 minutes, and then you continue walking at a leisurely pace of 1 mile per hour for 30 minutes."
find the domain of
40< t less than or equal to 70
To find the domain of the function within the given time frame, we need to determine the intervals during which the person is walking. In this case, the person walks for a total of 25 minutes at a fast pace, then rests for 15 minutes, and finally walks for another 30 minutes at a leisurely pace.
Let's define the function as follows:
D(t) = 6t for 0 < t ≤ 25
D(t) = 6(25) for 25 < t ≤ 40
D(t) = 6(25) for 40 < t ≤ 55
D(t) = 6(25) + 1(t-55) for 55 < t ≤ 70
Now, let's determine the domain of the function for the given time frame:
Since the given time frame is 40 < t ≤ 70, the person is walking at a leisurely pace for 30 minutes (from 40 to 70). Thus, the domain of the function is 40 ≤ t ≤ 70.
To find the domain of the function within the given time interval (40 < t <= 70), we need to determine the distance walked during this time.
The scenario can be divided into three parts:
1. Fast-paced walking for 25 minutes at 6 miles per hour: In this segment, the distance walked can be calculated using the formula d = rt, where d is the distance, r is the rate, and t is the time. So, the distance walked during this part is d1 = 6 * (25/60) miles.
2. Rest for 15 minutes: During this time, no distance is covered, so the equation for this segment is d2 = 0 miles.
3. Leisurely walking for 30 minutes at 1 mile per hour: Similarly, using the same formula, the distance covered during this part is d3 = 1 * (30/60) miles.
Now, combining the three segments, we have the piecewise function:
D(t) = { 6 * (t/60) miles for 0 ≤ t < 25
{ d1 + d2 = 6 * (25/60) miles for 25 ≤ t < 40
{ d1 + d2 + (t-40)/60 miles for 40 ≤ t ≤ 70
To find the domain within the time interval 40 < t <= 70, we need to check if there are any restrictions on t for each segment of the function.
In the first segment, when 0 ≤ t < 25, there are no restrictions since it covers the entire interval given.
In the second segment, when 25 ≤ t < 40, the domain is restricted to t values between 25 and 40, inclusive.
In the third segment, when 40 ≤ t ≤ 70, there are no restrictions since it covers the entire interval given.
Therefore, the domain of the function within the specified time interval is:
Domain: 25 ≤ t ≤ 40