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."
i have:
D(t)= (1/10)t if 0</t</30
D(t)= __?__ if 30 <t</40
D(t)= __?__ if 40<t</65
you have walked __?__ miles @ the end of 24 mins.
you have walked __?__ miles @ the end of 37 mins.
you have walked __?__ miles @ the end of 55 mins.
Based on the given scenario, we can define the piecewise function as follows:
D(t) = (1/10)t if 0 ≤ t ≤ 30
D(t) = 6 if 30 < t ≤ 40
D(t) = 6 + (1/60)(t - 40) if 40 < t ≤ 65
To find the distances at different time intervals, we substitute the values of t into the function:
1. At the end of 24 minutes:
D(24) = (1/10)(24) = 2.4 miles
2. At the end of 37 minutes:
D(37) = 6 (since it falls within the interval 30 < t ≤ 40) = 6 miles
3. At the end of 55 minutes:
D(55) = 6 + (1/60)(55 - 40)
= 6 + (1/60)(15)
= 6 + 0.25
= 6.25 miles
Therefore, at the end of 24 minutes, you have walked 2.4 miles. At the end of 37 minutes, you have walked 6 miles. At the end of 55 minutes, you have walked 6.25 miles.
To create the piecewise function, let's break down the scenario into three parts based on the given time intervals:
1. From 0 to 30 minutes (fast pace of 6 miles per hour):
D(t) = (1/10)t
2. From 30 to 40 minutes (resting for 10 minutes):
D(t) = The distance remains constant. (We will find the exact value later.)
3. From 40 to 65 minutes (leisurely pace of 1 mile per hour):
D(t) = The distance covered during the first 30 minutes plus the additional distance covered at the leisurely pace.
D(t) = (1/10)t + Additional distance.
To find the additional distance, we need to calculate the distance covered during the 25 minutes at the leisurely pace:
Distance = Speed * Time = 1 mile/hour * (t - 40) minutes.
Since t represents minutes, we need to convert 1 hour to 60 minutes.
Distance = (1 mile/hour) * (t - 40)/60.
Now, let's fill in the missing parts of the piecewise function:
D(t) = (1/10)t if 0 ≤ t ≤ 30
D(t) = ? if 30 < t ≤ 40
D(t) = (1/10)t + (1/60)(t - 40) if 40 < t ≤ 65
To find the distance at specific times, we can substitute these values into the function:
To find the distance @ the end of 24 mins:
D(24) = (1/10)(24) = 2.4 miles
To find the distance @ the end of 37 mins:
D(37) = ? (between 30 and 40 minutes, so we need to determine the constant value during that time)
To find the distance @ the end of 55 mins:
D(55) = (1/10)(55) + (1/60)(55 - 40) = 5.5 + 0.25 = 5.75 miles