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 3 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."
To create a piecewise function for this scenario, we need to consider the different intervals and rates at which you are walking.
Let's break it down step by step:
1. Fast pace: You walk at a rate of 3 miles per hour for 30 minutes. This means you walk a total distance of 3 * (30/60) = 1.5 miles during this period.
2. Rest period: You rest for 10 minutes, so the distance you walk during this time is 0.
3. Leisurely pace: You walk at a rate of 1 mile per hour for 25 minutes. Therefore, you walk a total distance of 1 * (25/60) = 0.4167 miles during this interval.
Now that we've calculated the distances for each interval, we can build the piecewise function.
Let's define the function as Upper D left-parenthesis t right-parenthesis, where t represents the time in minutes. The piecewise function will be as follows:
Upper D left-parenthesis t right-parenthesis =
{
3t/60, for 0 ≤ t ≤ 30 (fast pace)
1.5, for 30 < t ≤ 40 (rest period)
1.5 + (t - 40)/60, for 40 < t ≤ 65 (leisurely pace)
1.9167, for t > 65 (after the walk)
}
The function is broken into four intervals:
- From 0 to 30 minutes, the distance is given by 3t/60, representing the fast pace.
- From 30 to 40 minutes (inclusive), the distance is a constant 1.5 miles, representing the rest period.
- From 40 to 65 minutes (inclusive), the distance is given by 1.5 + (t - 40)/60, representing the leisurely pace.
- For any time greater than 65 minutes, the walk is complete, and the distance remains constant at 1.9167 miles (1.5 + 25/60).
That's the piecewise function representing the total distance walked based on the given scenario.