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 for this scenario, we can define the different time intervals and the corresponding places in the race. Let's denote the time completed as "t" and the place in the race as "P(t)".
Within each time interval, the place in the race will be constant. We can define the intervals based on the given information:
Interval 1: t ≥ 140 and t < 145 (for third place)
Interval 2: t ≥ 145 and t < 150 (for fourth place)
Interval 3: t ≥ 150 and t < 155 (for fifth place)
Interval 4: t ≥ 155 and t < 160 (for sixth place)
Interval 5: t ≥ 160 and t < 165 (for seventh place)
Interval 6: t ≥ 165 and t < 170 (for eighth place)
Interval 7: t ≥ 170 and t < 175 (for ninth place)
Interval 8: t ≥ 175 and t ≤ 168 (for tenth place)
Now, let's define the piecewise function:
P(t) = 3, for 140 ≤ t < 145
P(t) = 4, for 145 ≤ t < 150
P(t) = 5, for 150 ≤ t < 155
P(t) = 6, for 155 ≤ t < 160
P(t) = 7, for 160 ≤ t < 165
P(t) = 8, for 165 ≤ t < 170
P(t) = 9, for 170 ≤ t < 175
P(t) = 10, for 175 ≤ t ≤ 168
This piecewise function assigns the corresponding place in the race based on the given time completed.
To graphically represent this model, you can plot the time completed (t) on the x-axis and the place in the race (P(t)) on the y-axis. Since the function is discontinuous at each interval, you can represent each interval as a step function or a bar graph with constant values and breaks at the interval endpoints.
I hope this helps! Let me know if you have any further questions.
To create a piecewise function that gives your place in the race based on the time completed, we need to define different intervals for the consecutive places in the race.
First, let's assign place numbers to the corresponding times:
1st place: 130 minutes
2nd place: 135 minutes
3rd place: 140 minutes
...
10th place: 168 minutes
Since the contestants come in at regular intervals, we can deduce that each place time increases by 5 minutes. So, the nth place time can be calculated using the formula:
nth place time = (130 + (n-1)*5) minutes
Now we can represent the piecewise function using interval notation and create an algebraic and graphical model.
Let x represent the time completed in minutes.
For 130 ≤ x < 135, the given function will output 1 (first place). So, the algebraic model for this interval is:
f(x) = 1, for 130 ≤ x < 135
For 135 ≤ x < 140, the given function will output 2 (second place). So, the algebraic model for this interval is:
f(x) = 2, for 135 ≤ x < 140
Similarly, we can continue this pattern for the remaining places. The algebraic model for the nth place interval (140 ≤ x < 130 + (n-1)*5) is:
f(x) = n, for 140 ≤ x < 130 + (n-1)*5
Graphically, you can plot the piecewise function by representing time completed (x) on the x-axis and place in the race (f(x)) on the y-axis. You will have different horizontal segments representing different places in the race.
Note that this model assumes no ties between the contestants and that each place in the race is achieved by exactly one contestant.