At a certain marathon race, the winner comes in at a time of 130 minutes. Second place comes in at a time of 135 minutes, and third place comes in at 140 minutes. Starting with third place, contestants come in at regular intervals all the way up to the last successful finisher, who gets 10th place with a time of 168 minutes.
Using a piecewise function, create an algebraic and graphical model that would output your place in the race according to time completed.
To create a piecewise function, we need to define the different intervals for the places in the race.
Let's divide the intervals based on the time completed:
Interval 1: 130 ≤ time ≤ 135 (Winner)
Interval 2: 135 < time ≤ 140 (Second place)
Interval 3: 140 < time ≤ 145 (Third place)
Interval 4: 145 < time ≤ 150 (Fourth place)
Interval 5: 150 < time ≤ 155
Interval 6: 155 < time ≤ 160
Interval 7: 160 < time ≤ 165
Interval 8: 165 < time ≤ 168 (10th place)
Now, let's create the piecewise function:
- For 130 ≤ time ≤ 135, the output is 1 (Winner)
- For 135 < time ≤ 140, the output is 2 (Second place)
- For 140 < time ≤ 145, the output is 3 (Third place)
- For 145 < time ≤ 150, the output is 4 (Fourth place)
- For 150 < time ≤ 155, the output is 5
- For 155 < time ≤ 160, the output is 6
- For 160 < time ≤ 165, the output is 7
- For 165 < time ≤ 168, the output is 8 (10th place)
Algebraically, we can represent this as:
f(time) = {
1, 130 ≤ time ≤ 135
2, 135 < time ≤ 140
3, 140 < time ≤ 145
4, 145 < time ≤ 150
5, 150 < time ≤ 155
6, 155 < time ≤ 160
7, 160 < time ≤ 165
8, 165 < time ≤ 168
}
Graphically, the function can be represented as a step function, where the y-axis represents the place in the race and the x-axis represents the time completed. Here is a graphical representation of the function:
9
|
|
| 8
|
| 7
Place |
in Race |
| 6
|
|
| 5
|
|
| 4
|
|
| 3
|
|
| 2
|
|
|
_________|__________________________________________________________________________
130 135 140 145 150 155 160 165 168
time completed
To create a piecewise function that represents the relationship between time completed and place in the race, we need to define different intervals and their corresponding place values.
Let's break down the intervals:
Interval 1: 130 minutes to 135 minutes
- Since the winner finished at 130 minutes, this interval only includes the second-place finisher.
- In this interval, the place value is 2.
Interval 2: 135 minutes to 140 minutes
- This interval includes the third-place finisher.
- In this interval, the place value is 3.
Interval 3: 140 minutes to 145 minutes
- This interval includes the fourth-place finisher.
- In this interval, the place value is 4.
Interval 4: 145 minutes to 150 minutes
- This interval includes the fifth-place finisher.
- In this interval, the place value is 5.
Interval 5: 150 minutes to 155 minutes
- This interval includes the sixth-place finisher.
- In this interval, the place value is 6.
Interval 6: 155 minutes to 160 minutes
- This interval includes the seventh-place finisher.
- In this interval, the place value is 7.
Interval 7: 160 minutes to 165 minutes
- This interval includes the eighth-place finisher.
- In this interval, the place value is 8.
Interval 8: 165 minutes to 168 minutes
- This interval includes the ninth and tenth-place finishers.
- In this interval, the place value is 9.
Now, let's represent this information using a piecewise function:
f(x) =
2, 130 ≤ x < 135
3, 135 ≤ x < 140
4, 140 ≤ x < 145
5, 145 ≤ x < 150
6, 150 ≤ x < 155
7, 155 ≤ x < 160
8, 160 ≤ x < 165
9, 165 ≤ x < 168
10, 168 ≤ x
This piecewise function means that if you input a time completed (x), the function will output your place in the race.
To graph this piecewise function, you can plot the intervals along the x-axis and the corresponding place values along the y-axis. Each interval will be represented by a horizontal line segment at the respective place value.
I hope this explanation helps! If you have any further questions, please let me know.