Question

julissa is running a 10 - kilometer race at a constant pace. after running for 18 minutes, she completes 2 kilometers. after running for 54 minutes, she completes 6 kilometers. her trainer writes an equation letting t, the time in minutes, represent the independent variable and k, the number of kilometers, represent the dependent variable. which equation can be used to represent k, the number of kilometers julissa runs in t minutes? \\(\bigcirc\\) \\(k - 2 = \frac{1}{9}(t - 18)\\) \\(\bigcirc\\) \\(k - 18 = \frac{1}{9}(t - 2)\\) \\(\bigcirc\\) \\(k - 2 = 9(t - 18)\\) \\(\bigcirc\\) \\(k - 18 = 9(t - 2)\\)

Explanation:

Step1: Find the running rate

First, we calculate the rate at which Julissa runs. We know that after 18 minutes, she runs 2 kilometers, and after 54 minutes, she runs 6 kilometers. The rate \( r \) (in kilometers per minute) can be calculated as the change in distance over the change in time. So, \( r=\frac{6 - 2}{54 - 18}=\frac{4}{36}=\frac{1}{9}\) km per minute? Wait, no, wait. Wait, actually, let's check with the first two points: at \( t = 18\), \( k=2\); at \( t = 54\), \( k = 6\). The slope (rate) is \(\frac{6 - 2}{54 - 18}=\frac{4}{36}=\frac{1}{9}\)? Wait, no, \(\frac{6 - 2}{54 - 18}=\frac{4}{36}=\frac{1}{9}\)? Wait, no, 4 divided by 36 is \(\frac{1}{9}\)? Wait, no, 4/36 simplifies to 1/9? Wait, 4 divided by 36: divide numerator and denominator by 4, we get 1/9. Wait, but also, if we take the first point: at \( t = 18\), \( k = 2\). So the equation of the line in point - slope form is \(k - k_1=m(t - t_1)\), where \(m\) is the slope and \((t_1,k_1)\) is a point on the line.

We can also calculate the rate as \(\frac{2}{18}=\frac{1}{9}\) km per minute (since in 18 minutes, she runs 2 km) and \(\frac{6}{54}=\frac{1}{9}\) km per minute (in 54 minutes, 6 km). So the rate \(m=\frac{1}{9}\) km per minute.

Using the point - slope form with the point \((t_1 = 18,k_1 = 2)\), the equation is \(k - 2=\frac{1}{9}(t - 18)\).

Let's verify with the other point. If \(t = 54\), then \(k-2=\frac{1}{9}(54 - 18)=\frac{1}{9}\times36 = 4\), so \(k=4 + 2=6\), which matches the given information (after 54 minutes, she runs 6 km).

Let's check the other options:

• Option 2: \(k - 18=\frac{1}{9}(t - 2)\). If \(t = 18\), then \(k-18=\frac{1}{9}(16)\), \(k=18+\frac{16}{9}\approx19.78\), which is not 2. So this is wrong.

• Option 3: \(k - 2=9(t - 18)\). If \(t = 54\), \(k-2=9\times36=324\), \(k = 326\), which is wrong.

• Option 4: \(k - 18=9(t - 2)\). If \(t = 18\), \(k-18=9\times16 = 144\), \(k=162\), which is wrong.

Answer:

\(k - 2=\frac{1}{9}(t - 18)\) (the first option in the list, i.e., the option with the equation \(k - 2=\frac{1}{9}(t - 18)\))