Sure, let’s go through a detailed, step-by-step solution for each of the three questions based on the provided dataset.
We are given the following data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (minutes)} & \text{Total Distance (km)} \\
\hline
0 & 0 \\
\hline
30 & 6.8 \\
\hline
60 & 10.1 \\
\hline
90 & 12 \\
\hline
120 & 13.3 \\
\hline
150 & 15 \\
\hline
\end{array}
\][/tex]
### 1. What is the dependent variable?
In our dataset, the dependent variable is the one that depends on the other variable. Here, the total distance run by Natalie changes depending on how many minutes she has been running. Hence, the dependent variable is:
[tex]\[
\text{Total Distance}
\][/tex]
### 2. What is the independent variable?
The independent variable is the one that influences or determines the values of other variables. In this context, the total distance run is measured at different points in time. Thus, the independent variable is:
[tex]\[
\text{Time}
\][/tex]
### 3. How many kilometers had Natalie run after 40 minutes?
To find out how many kilometers Natalie had run after 40 minutes, we can estimate using linear interpolation between the two given time points that surround 40 minutes. From the data:
- At 30 minutes, Natalie had run 6.8 kilometers.
- At 60 minutes, Natalie had run 10.1 kilometers.
We need to estimate the distance at 40 minutes, which lies between 30 and 60 minutes. The formula for linear interpolation is given by:
[tex]\[
d = d_1 + \left( \frac{d_2 - d_1}{t_2 - t_1} \right) \times (t - t_1)
\][/tex]
where:
- [tex]\(d\)[/tex] is the distance at the desired time (40 minutes).
- [tex]\(d_1\)[/tex] is the distance at the previous known time (30 minutes, which is 6.8 km).
- [tex]\(d_2\)[/tex] is the distance at the next known time (60 minutes, which is 10.1 km).
- [tex]\(t_1\)[/tex] is the previous known time (30 minutes).
- [tex]\(t_2\)[/tex] is the next known time (60 minutes).
- [tex]\(t\)[/tex] is the desired time (40 minutes).
Substitute the values:
[tex]\[
d = 6.8 + \left( \frac{10.1 - 6.8}{60 - 30} \right) \times (40 - 30)
\][/tex]
[tex]\[
d = 6.8 + \left( \frac{3.3}{30} \right) \times 10
\][/tex]
[tex]\[
d = 6.8 + 1.1
\][/tex]
[tex]\[
d = 7.9
\][/tex]
Therefore, after 40 minutes, Natalie had run:
[tex]\[
7.9 \text{ kilometers}
\][/tex]
