Given the data points, we need to determine the correct equation to represent the number of kilometers [tex]\( k \)[/tex] Julissa runs in [tex]\( t \)[/tex] minutes.
We know the following:
- After 18 minutes, Julissa has run 2 kilometers.
- After 54 minutes, Julissa has run 6 kilometers.
First, let's determine the rate at which she is running. The formula for the rate (slope) is:
[tex]\[
\text{slope} = \frac{\Delta k}{\Delta t} = \frac{k_2 - k_1}{t_2 - t_1}
\][/tex]
Plugging in the given values:
[tex]\[
\text{slope} = \frac{6 - 2}{54 - 18} = \frac{4}{36} = \frac{1}{9}
\][/tex]
Now, using the point-slope form of the equation of a line, which is:
[tex]\[
k - k_1 = \text{slope} \times (t - t_1)
\][/tex]
We can choose either of the points [tex]\((18, 2)\)[/tex] or [tex]\((54, 6)\)[/tex]. Let's use [tex]\((18, 2)\)[/tex] to derive the equation:
[tex]\[
k - 2 = \frac{1}{9} \times (t - 18)
\][/tex]
Thus, the equation that represents [tex]\( k \)[/tex] is:
[tex]\[
k - 2 = \frac{1}{9}(t - 18)
\][/tex]
Therefore, the correct equation is:
[tex]\[
k - 2 = \frac{1}{9}(t - 18)
\][/tex]
