Ann's ride , a piecewise function
Ann went on a 10 hour bicycle trip. The table shows selected points on the graph.
t-0-1-4-7-10
d (t)- 0-3-5-5-10.
Write a function d(t) for her total distance traveled in miles in terms of the time in hours. Please help!
Well, it will surely be a piecewise function, so
d(t) =
3t for 0<=t<=1
3+2(t-1)/3 for 1<t<=4
5 for 4<t<=7
5+5(t-7)/3
Just look at the slope of each piece, and add the offset where it starts.
Sure, here's my attempt at a humorous answer:
Well, it looks like Ann's ride is like a roller coaster of emotions, or in this case, distances! Let's break it down:
For the first hour, Ann covered a grand total of 0 miles. She might as well have been riding on a stationary bike!
But then, in the second hour, she kicked things up a notch and traveled a distance of 3 miles. It's like she suddenly remembered she's not in a Tour de France race!
Then, in hours 3 to 6, Ann's progress seems to have hit a plateau. She maintained a constant speed and covered 5 miles each hour. Talk about finding your comfort zone!
But hold on, in the final hour, Ann went all out and reached a distance of 10 miles! She must have channeled her inner Olympic cyclist.
So, let's piece this all together into a function. We'll express the distance, d(t), where t is the time in hours:
d(t) = 0 for t ≤ 0,
d(t) = 3 for 0 < t ≤ 1,
d(t) = 5 for 1 < t ≤ 4,
d(t) = 5 for 4 < t ≤ 7,
d(t) = 10 for 7 < t ≤ 10.
So, there you have it. Ann's ride can be charted using this hilarious piecewise function. Enjoy the ups and downs (literally)!
To write a function for Ann's total distance traveled in terms of time, we need to consider the different intervals in the table.
For t = 0 to t = 1, the distance traveled is 3 - 0 = 3 miles.
For t = 1 to t = 4, the distance traveled is 5 - 3 = 2 miles.
For t = 4 to t = 7, the distance traveled is 5 - 5 = 0 miles.
For t = 7 to t = 10, the distance traveled is 10 - 5 = 5 miles.
Based on these intervals, we can write the piecewise function for the total distance traveled as follows:
d(t) = 3, for 0 ≤ t < 1
d(t) = 3 + 2(t - 1) = 2t + 1, for 1 ≤ t < 4
d(t) = 5, for 4 ≤ t < 7
d(t) = 5 + 5(t - 7) = 5t - 30, for 7 ≤ t ≤ 10
So, the function d(t) for her total distance traveled in terms of time in hours is:
d(t) = 3, for 0 ≤ t < 1
d(t) = 2t + 1, for 1 ≤ t < 4
d(t) = 5, for 4 ≤ t < 7
d(t) = 5t - 30, for 7 ≤ t ≤ 10
To write a function for Ann's total distance traveled in terms of time, we can observe the given data points and identify the pattern.
First, let's consider the time intervals between the given points. From t=0 to t=1, Ann traveled a distance of 3 miles (d=3-0=3). From t=1 to t=4, Ann traveled a distance of 2 miles (d=5-3=2). From t=4 to t=7, Ann traveled a distance of 0 miles (d=5-5=0). Finally, from t=7 to t=10, Ann traveled a distance of 5 miles (d=10-5=5).
We can observe that the distance traveled changes at specific time points and remains constant until the next time point.
Based on this observation, we can divide the 10-hour interval into multiple segments and define the distance traveled for each segment. Let's consider:
- For t between 0 and 1, the distance traveled is 3t. (d=3t)
- For t between 1 and 4, the distance traveled is 3 + 2(t-1). (d=3+2(t-1))
- For t between 4 and 7, the distance traveled is 5. (d=5)
- For t between 7 and 10, the distance traveled is 5 + 5(t-7). (d=5+5(t-7))
Now we can define the piecewise function d(t) as follows:
d(t) = {
3t, for 0 <= t < 1,
3 + 2(t-1), for 1 <= t < 4,
5, for 4 <= t < 7,
5 + 5(t-7), for 7 <= t <= 10
}
Therefore, the function for Ann's total distance traveled in terms of time is:
d(t) = {
3t, for 0 <= t < 1,
3 + 2(t-1), for 1 <= t < 4,
5, for 4 <= t < 7,
5 + 5(t-7), for 7 <= t <= 10
}
Note: Since the time interval is given as continuous, we assume that the function is continuous and the distance traveled smoothly changes between the given time points.