To determine which equation represents the relationship between the distance a car travels, [tex]\(y\)[/tex], in miles, and the time [tex]\(x\)[/tex], in minutes, we'll analyze the data given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time, } x & \text{Distance, } y \\
\hline
48 & 12 \\
64 & 16 \\
72 & 18 \\
\hline
\end{array}
\][/tex]
First, we calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair of values:
1. For [tex]\(x = 48\)[/tex] and [tex]\(y = 12\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{12}{48} = \frac{1}{4}
\][/tex]
2. For [tex]\(x = 64\)[/tex] and [tex]\(y = 16\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{16}{64} = \frac{1}{4}
\][/tex]
3. For [tex]\(x = 72\)[/tex] and [tex]\(y = 18\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{18}{72} = \frac{1}{4}
\][/tex]
We observe that the ratio [tex]\( \frac{y}{x} \)[/tex] is consistent for all data points and equals [tex]\( \frac{1}{4} \)[/tex]. This implies a linear relationship between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] and can be expressed as:
[tex]\[
y = \frac{1}{4} x
\][/tex]
Therefore, the equation that represents the relationship between the distance [tex]\(y\)[/tex] and the time [tex]\(x\)[/tex] for the car is:
[tex]\[
y = \frac{1}{4} x
\][/tex]
