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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 per 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]\[
\begin{tabular}{|c|c|c|}
\hline
\begin{tabular}{c}
$x$ \\
(seconds)
\end{tabular}
&
\begin{tabular}{c}
Zeke's \\
Distance \\
(yards)
\end{tabular}
&
\begin{tabular}{c}
Niko's \\
Distance \\
(yards)
\end{tabular} \\
\hline
0 & 0 & 12 \\
\hline
2 & 4 & 13 \\
\hline
6 & 12 & 15 \\
\hline
\end{tabular}
\][/tex]

(b) Based on your calculations for (a), write equations for both Zeke's distance and Niko's distance from the start line as a function of time, [tex]$x$[/tex].

- Zeke's Distance: [tex] y = 2x [/tex]
- Niko's Distance: [tex] y = \frac{1}{2}x + 12 [/tex]

(c) Graph both of these equations and determine the number of seconds it takes for Zeke to catch up to Niko.

(d) How far are they from the finish line at that point?

Answer :

Sure! Let's break down the solution to this problem step-by-step:

(a) Fill out the table for various distances:

For each time [tex]\(x\)[/tex], we calculate the distances for Zeke and Niko.

- For [tex]\(x = 0\)[/tex] seconds:
- Zeke's Distance: [tex]\(2 \times 0 = 0\)[/tex] yards
- Niko's Distance: [tex]\(12 + 0.5 \times 0 = 12\)[/tex] yards

- For [tex]\(x = 2\)[/tex] seconds:
- Zeke's Distance: [tex]\(2 \times 2 = 4\)[/tex] yards
- Niko's Distance: [tex]\(12 + 0.5 \times 2 = 13\)[/tex] yards

- For [tex]\(x = 6\)[/tex] seconds:
- Zeke's Distance: [tex]\(2 \times 6 = 12\)[/tex] yards
- Niko's Distance: [tex]\(12 + 0.5 \times 6 = 15\)[/tex] yards

Here's the completed table:

[tex]\[
\begin{array}{|c|c|c|}
\hline
x\,(\text{sec}) & \text{Zeke's Distance (yds)} & \text{Niko's Distance (yds)} \\
\hline
0 & 0 & 12 \\
2 & 4 & 13 \\
6 & 12 & 15 \\
\hline
\end{array}
\][/tex]

(b) Write equations for their distances:

- Zeke runs 2 yards every second, so his distance equation based on time [tex]\(x\)[/tex] is:
[tex]\[
\text{Zeke's Distance} = 2x
\][/tex]

- Niko gets a 12-yard head start and runs 1 yard every 2 seconds (or 0.5 yards per second), so his distance equation is:
[tex]\[
\text{Niko's Distance} = 12 + 0.5x
\][/tex]

(c) Determine when Zeke catches up to Niko:

To find when Zeke catches Niko, we set their distance equations equal to each other:

[tex]\[
2x = 12 + 0.5x
\][/tex]

Solving for [tex]\(x\)[/tex]:

1. Subtract [tex]\(0.5x\)[/tex] from both sides: [tex]\(2x - 0.5x = 12\)[/tex]
2. [tex]\(1.5x = 12\)[/tex]
3. Divide both sides by 1.5: [tex]\(x = \frac{12}{1.5} = 8\)[/tex] seconds

So, it takes 8 seconds for Zeke to catch up to Niko.

(d) Determine how far they are from the finish line:

At the time Zeke catches Niko, we find their distance from the start line using the equation [tex]\(2x\)[/tex]:

- Distance from start line for Zeke (or Niko, since they catch up): [tex]\(2 \times 8 = 16\)[/tex] yards

The race is 30 yards long, so the distance from the finish line is:

[tex]\[
30 - 16 = 14\text{ yards}
\][/tex]

Therefore, both Zeke and Niko are 14 yards from the finish line when Zeke catches up.

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