Create a piecewise function of the following scenario where Upper D left-parenthesis t right-parenthesis 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 4 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 30 minutes.

why all the words? Just say D(t).

I'll do the first piece. See what you can do with the rest.

Starting from t=0, having gone 0 miles so far,
D(t) = 0 + 4/60 (t-0) for 0 <= t <= 30
= 1/15 t

That may seem cumbersome, but therein lie hints for the other parts.

where did you get the 4 over 60

t is in minutes, and D is in miles.

Since the speed is given in mi/hr, you have to divide that by 60 to get mi/min

Thank You Steve I finally understand

To create a piecewise function for this scenario, we need to break down the different time intervals and the corresponding distances walked. Let's assign variables to each time interval:

t1: Time spent on fast pace (4 miles per hour) - 30 minutes
t2: Time spent resting - 10 minutes
t3: Time spent on leisurely pace (1 mile per hour) - 30 minutes

The first interval is when you walk at a fast pace for 30 minutes. At 4 miles per hour, the distance covered can be calculated by multiplying the pace by the time: 4 * (30/60) = 2 miles.

The second interval is when you rest for 10 minutes. No distance is covered during this time, so the distance will be 0.

The third interval is when you walk at a leisurely pace of 1 mile per hour for 30 minutes. In this case, we multiply the pace by the time to get the distance: 1 * (30/60) = 0.5 miles.

Putting it all together, we can define the piecewise function:

D(t) = { 2, t ≤ t1
{ 2, t1 < t ≤ (t1 + t2)
{ 2 + 0.5(t - t1 - t2), t > (t1 + t2)

To summarize:
1. For the first 30 minutes (t ≤ 30), the total distance walked is 2 miles.
2. After 30 minutes and when 30 < t ≤ 40 (adding the resting time), the total distance remains 2 miles.
3. For t > 40, the total distance is given by 2 + 0.5(t - 40).