At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
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.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
