At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. 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.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
