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 that outputs the place in the race according to the time completed, we can define different intervals for each place and use step functions.

Let's say 'x' represents the time completed (in minutes) and 'P(x)' represents the place in the race.

The intervals can be defined as follows:

For x > 168: No successful finisher, so P(x) = N/A
For 168 ≥ x > 163: 10th place, so P(x) = 10
For 163 ≥ x > 158: 9th place, so P(x) = 9
For 158 ≥ x > 153: 8th place, so P(x) = 8
For 153 ≥ x > 148: 7th place, so P(x) = 7
For 148 ≥ x > 143: 6th place, so P(x) = 6
For 143 ≥ x > 138: 5th place, so P(x) = 5
For 138 ≥ x > 133: 4th place, so P(x) = 4
For 133 ≥ x > 130: 3rd place, so P(x) = 3
For 130 ≥ x > 135: 2nd place, so P(x) = 2
For 135 ≥ x ≥ 140: 1st place, so P(x) = 1

Graphically, this can be represented as a step function with different constant values within each interval. The x-axis represents time completed (x) and the y-axis represents place in the race (P(x)).

Here is a graphical representation of the piecewise function:


| 1 |
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| 4 |
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| 10 |
+-----+
140 150 160 170

Note: The vertical lines are for visualization purposes only and should not be included in the actual graph.

This graph shows how the place in the race changes depending on the time completed.