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Zeke is racing his little brother Niko in a 30-yard race. Zeke gives Niko a 12-yard head start. Zeke runs 2 yards every second, while Niko runs 1 yard every 2 seconds. If [tex]$x$[/tex] represents the number of seconds they have been running and [tex]$y$[/tex] represents the distance from the start line, then:

(a) Fill out the table below for various distances that Zeke and Niko are from the start line at the given times.

[tex]
\[
\begin{array}{|c|c|c|}
\hline
x & \text{Zeke's Distance (yds)} & \text{Niko's Distance (yds)} \\
\hline
0 & & \\
1 & & \\
2 & & \\
3 & & \\
4 & & \\
\hline
\end{array}
\]
[/tex]

Answer :

Let's solve the problem step-by-step to find the distances that Zeke and Niko are from the start line at different times:

### Understanding the Problem
- Zeke runs 2 yards every second.
- Niko has a 12-yard head start and runs 1 yard every 2 seconds.

### Distance Formulas
- Zeke's Distance Formula: Since he runs 2 yards every second, after [tex]\(x\)[/tex] seconds, his distance from the start is:
[tex]\[
y_{\text{Zeke}} = 2x
\][/tex]
- Niko's Distance Formula: He starts 12 yards ahead and runs 1 yard every 2 seconds, which is 0.5 yards per second. Therefore, after [tex]\(x\)[/tex] seconds, his distance from the start is:
[tex]\[
y_{\text{Niko}} = 12 + 0.5x
\][/tex]

### Filling Out the Table

Let's calculate the distances for both Zeke and Niko from the start line at different times:

- At 0 seconds:
- Zeke: [tex]\(2 \times 0 = 0\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 0 = 12\)[/tex] yards

- At 1 second:
- Zeke: [tex]\(2 \times 1 = 2\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 1 = 12.5\)[/tex] yards

- At 2 seconds:
- Zeke: [tex]\(2 \times 2 = 4\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 2 = 13\)[/tex] yards

- At 3 seconds:
- Zeke: [tex]\(2 \times 3 = 6\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 3 = 13.5\)[/tex] yards

- At 4 seconds:
- Zeke: [tex]\(2 \times 4 = 8\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 4 = 14\)[/tex] yards

- At 5 seconds:
- Zeke: [tex]\(2 \times 5 = 10\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 5 = 14.5\)[/tex] yards

- At 6 seconds:
- Zeke: [tex]\(2 \times 6 = 12\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 6 = 15\)[/tex] yards

- At 7 seconds:
- Zeke: [tex]\(2 \times 7 = 14\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 7 = 15.5\)[/tex] yards

- At 8 seconds:
- Zeke: [tex]\(2 \times 8 = 16\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 8 = 16\)[/tex] yards

- At 9 seconds:
- Zeke: [tex]\(2 \times 9 = 18\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 9 = 16.5\)[/tex] yards

- At 10 seconds:
- Zeke: [tex]\(2 \times 10 = 20\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 10 = 17\)[/tex] yards

- At 11 seconds:
- Zeke: [tex]\(2 \times 11 = 22\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 11 = 17.5\)[/tex] yards

- At 12 seconds:
- Zeke: [tex]\(2 \times 12 = 24\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 12 = 18\)[/tex] yards

- At 13 seconds:
- Zeke: [tex]\(2 \times 13 = 26\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 13 = 18.5\)[/tex] yards

- At 14 seconds:
- Zeke: [tex]\(2 \times 14 = 28\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 14 = 19\)[/tex] yards

- At 15 seconds:
- Zeke: [tex]\(2 \times 15 = 30\)[/tex] yards
- Niko: [tex]\(12 + 0.5 \times 15 = 19.5\)[/tex] yards

These calculations give you the distances both Zeke and Niko are from the starting point at each second.

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