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Sagot :
To determine the rate of change of the distance Carol traveled while cross-country skiing, we need to calculate the rate of change for each consecutive pair of points (minutes, distance). Let's break it down step-by-step.
### Step-by-Step Solution:
1. Identify Consecutive Pairs of Points:
We have the following data points:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes} & \text{Distance Traveled (miles)} \\ \hline 2 & \frac{1}{6} \\ \hline 3 & \frac{17}{48} \\ \hline 4 & \frac{13}{24} \\ \hline 5 & \frac{35}{48} \\ \hline 6 & \frac{11}{12} \\ \hline \end{array} \][/tex]
2. Calculate Rate of Change for Each Consecutive Pair:
The rate of change between two points [tex]\((t_1, d_1)\)[/tex] and [tex]\((t_2, d_2)\)[/tex] is given by:
[tex]\[ \text{Rate of Change} = \frac{d_2 - d_1}{t_2 - t_1} \][/tex]
- Between (2, [tex]\(\frac{1}{6}\)[/tex]) and (3, [tex]\(\frac{17}{48}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{17}{48} - \frac{1}{6}}{3 - 2} = \frac{\frac{17}{48} - \frac{8}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (3, [tex]\(\frac{17}{48}\)[/tex]) and (4, [tex]\(\frac{13}{24}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{13}{24} - \frac{17}{48}}{4 - 3} = \frac{\frac{26}{48} - \frac{17}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (4, [tex]\(\frac{13}{24}\)[/tex]) and (5, [tex]\(\frac{35}{48}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{35}{48} - \frac{13}{24}}{5 - 4} = \frac{\frac{35}{48} - \frac{26}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (5, [tex]\(\frac{35}{48}\)[/tex]) and (6, [tex]\(\frac{11}{12}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{11}{12} - \frac{35}{48}}{6 - 5} = \frac{\frac{44}{48} - \frac{35}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
3. Summarize the Rates of Change:
The calculated rates of change between each consecutive pair of points are:
[tex]\[ [0.1875, 0.1875, 0.1875, 0.1875] \][/tex]
Hence, the consistent rate of change of the distance Carol traveled while cross-country skiing is approximately [tex]\(0.1875\)[/tex] miles per minute.
### Step-by-Step Solution:
1. Identify Consecutive Pairs of Points:
We have the following data points:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes} & \text{Distance Traveled (miles)} \\ \hline 2 & \frac{1}{6} \\ \hline 3 & \frac{17}{48} \\ \hline 4 & \frac{13}{24} \\ \hline 5 & \frac{35}{48} \\ \hline 6 & \frac{11}{12} \\ \hline \end{array} \][/tex]
2. Calculate Rate of Change for Each Consecutive Pair:
The rate of change between two points [tex]\((t_1, d_1)\)[/tex] and [tex]\((t_2, d_2)\)[/tex] is given by:
[tex]\[ \text{Rate of Change} = \frac{d_2 - d_1}{t_2 - t_1} \][/tex]
- Between (2, [tex]\(\frac{1}{6}\)[/tex]) and (3, [tex]\(\frac{17}{48}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{17}{48} - \frac{1}{6}}{3 - 2} = \frac{\frac{17}{48} - \frac{8}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (3, [tex]\(\frac{17}{48}\)[/tex]) and (4, [tex]\(\frac{13}{24}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{13}{24} - \frac{17}{48}}{4 - 3} = \frac{\frac{26}{48} - \frac{17}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (4, [tex]\(\frac{13}{24}\)[/tex]) and (5, [tex]\(\frac{35}{48}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{35}{48} - \frac{13}{24}}{5 - 4} = \frac{\frac{35}{48} - \frac{26}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
- Between (5, [tex]\(\frac{35}{48}\)[/tex]) and (6, [tex]\(\frac{11}{12}\)[/tex]):
[tex]\[ \text{Rate of Change} = \frac{\frac{11}{12} - \frac{35}{48}}{6 - 5} = \frac{\frac{44}{48} - \frac{35}{48}}{1} = \frac{\frac{9}{48}}{1} = \frac{9}{48} = 0.1875 \][/tex]
3. Summarize the Rates of Change:
The calculated rates of change between each consecutive pair of points are:
[tex]\[ [0.1875, 0.1875, 0.1875, 0.1875] \][/tex]
Hence, the consistent rate of change of the distance Carol traveled while cross-country skiing is approximately [tex]\(0.1875\)[/tex] miles per minute.
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