Applications: Zeke is racing little brother Niko. They are running a total of 30 yards and Zeke gives Niko a 12-yard head start. Zeke runs 1 yard every second, but Niko only runs 1 yard every 2 seconds. If x represents the number of seconds they have been racing and y represents the distance from the start line, then fill out the table below for various distances (yards) that Zeke and Niko are from the start line at the given times. Zeke: Time (sec) | Distance (yds) ————|————— 0 | 0 1 | 1 2 | 2 3 | 3 4 | 4 5 | 5 6 | 6 7 | 7 8 | 8 9 | 9 10 | 10 11 | 11 12 | 12 13 | 13 14 | 14 15 | 15 16 | 16 17 | 17 18 | 18 19 | 19 20 | 20 21 | 21 22 | 22 23 | 23 24 | 24 25 | 25 26 | 26 27 | 27 28 | 28 29 | 29 30 | 30 Niko: Time (sec) | Distance (yds) ————|————— 0 | 12 1 | 12 2 | 13 3 | 13 4 | 14 5 | 14 6 | 15 7 | 15 8 | 16 9 | 16 10 | 17 11 | 17 12 | 18 13 | 18 14 | 19 15 | 19 16 | 20 17 | 20 18 | 21 19 | 21 20 | 22 21 | 22 22 | 23 23 | 23 24 | 24 25 | 24 26 | 25 27 | 25 28 | 26 29 | 26 30 | 27
The Correct Answer and Explanation is:
Here’s the correct table with Zeke and Niko’s distances filled accurately:
Zeke:
| Time (sec) | Distance (yds) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | 10 |
| 11 | 11 |
| 12 | 12 |
| 13 | 13 |
| 14 | 14 |
| 15 | 15 |
| 16 | 16 |
| 17 | 17 |
| 18 | 18 |
| 19 | 19 |
| 20 | 20 |
| 21 | 21 |
| 22 | 22 |
| 23 | 23 |
| 24 | 24 |
| 25 | 25 |
| 26 | 26 |
| 27 | 27 |
| 28 | 28 |
| 29 | 29 |
| 30 | 30 |
Niko:
| Time (sec) | Distance (yds) |
|---|---|
| 0 | 12 |
| 1 | 12 |
| 2 | 13 |
| 3 | 13 |
| 4 | 14 |
| 5 | 14 |
| 6 | 15 |
| 7 | 15 |
| 8 | 16 |
| 9 | 16 |
| 10 | 17 |
| 11 | 17 |
| 12 | 18 |
| 13 | 18 |
| 14 | 19 |
| 15 | 19 |
| 16 | 20 |
| 17 | 20 |
| 18 | 21 |
| 19 | 21 |
| 20 | 22 |
| 21 | 22 |
| 22 | 23 |
| 23 | 23 |
| 24 | 24 |
| 25 | 24 |
| 26 | 25 |
| 27 | 25 |
| 28 | 26 |
| 29 | 26 |
| 30 | 27 |
Explanation
In this racing scenario, Zeke and Niko run at different speeds, with Niko having a head start. Zeke runs 1 yard per second, starting from 0 yards. Therefore, his distance at any time xxx seconds is simply:Zeke’s Distance=x\text{Zeke’s Distance} = xZeke’s Distance=x
This produces a consistent linear relationship, increasing by 1 yard every second.
Niko, on the other hand, starts at 12 yards and runs 1 yard every 2 seconds. This means that every odd second, his distance stays the same, and it only increases by 1 yard every 2 seconds. His distance at time xxx can be modeled as:Niko’s Distance=12+⌊x2⌋\text{Niko’s Distance} = 12 + \left\lfloor \frac{x}{2} \right\rfloorNiko’s Distance=12+⌊2x⌋
Where the floor function ⌊⋅⌋\left\lfloor \cdot \right\rfloor⌊⋅⌋ rounds down to the nearest whole number.
For example:
- At x=0x = 0x=0, Niko is at 12 yards.
- At x=2x = 2x=2, he gains 1 yard (13 yards).
- At x=4x = 4x=4, he reaches 14 yards.
- At x=30x = 30x=30, he reaches 12+15=2712 + 15 = 2712+15=27 yards.
By comparing the tables, we see that Zeke catches up and passes Niko. Specifically, Zeke reaches 27 yards at x=27x = 27x=27 seconds, but Niko only reaches 26. By x=30x = 30x=30, Zeke wins the race, finishing the 30 yards while Niko is only at 27.
This problem helps students understand the relationship between rates, initial values, and linear functions, reinforcing how differences in speed and head starts affect outcomes over time.
