To find the rate of change of the distance Carol traveled while cross-country skiing, we need to look at the change in distance traveled over the change in time. This can be expressed as:
\[ \text{Rate of Change} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{\Delta d}{\Delta t} \]
From the data provided, we will first calculate the distance traveled at each key point in time (in minutes):
- At 2 minutes: \(\frac{1}{6}\) miles
- At 3 minutes: \(\frac{17}{48}\) miles
- At 4 minutes: \(\frac{13}{24}\) miles
- At 5 minutes: \(\frac{35}{48}\) miles
- At 6 minutes: \(\frac{11}{12}\) miles
Next, let's analyze the change in distance traveled between each pair of consecutive time intervals.
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From 2 to 3 minutes: \[ \text{Change in Distance} = \frac{17}{48} - \frac{1}{6} = \frac{17}{48} - \frac{8}{48} = \frac{9}{48} = \frac{3}{16} \] \[ \text{Change in Time} = 3 - 2 = 1 \text{ minute} \] \[ \text{Rate of Change} = \frac{\frac{3}{16}}{1} = \frac{3}{16} \text{ miles per minute} \]
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From 3 to 4 minutes: \[ \text{Change in Distance} = \frac{13}{24} - \frac{17}{48} \] First, convert \(\frac{13}{24}\) to a common denominator of 48: \[ \frac{13}{24} = \frac{26}{48} \] Now calculate the change: \[ \text{Change in Distance} = \frac{26}{48} - \frac{17}{48} = \frac{9}{48} = \frac{3}{16} \] \[ \text{Change in Time} = 4 - 3 = 1 \text{ minute} \] \[ \text{Rate of Change} = \frac{\frac{3}{16}}{1} = \frac{3}{16} \text{ miles per minute} \]
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From 4 to 5 minutes: \[ \text{Change in Distance} = \frac{35}{48} - \frac{13}{24} \] Convert \(\frac{13}{24}\) again: \[ \text{Change in Distance} = \frac{35}{48} - \frac{26}{48} = \frac{9}{48} = \frac{3}{16} \] \[ \text{Change in Time} = 5 - 4 = 1 \text{ minute} \] \[ \text{Rate of Change} = \frac{\frac{3}{16}}{1} = \frac{3}{16} \text{ miles per minute} \]
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From 5 to 6 minutes: \[ \text{Change in Distance} = \frac{11}{12} - \frac{35}{48} \] Convert \(\frac{11}{12}\) to a common denominator of 48: \[ \frac{11}{12} = \frac{44}{48} \] Now calculate: \[ \text{Change in Distance} = \frac{44}{48} - \frac{35}{48} = \frac{9}{48} = \frac{3}{16} \] \[ \text{Change in Time} = 6 - 5 = 1 \text{ minute} \] \[ \text{Rate of Change} = \frac{\frac{3}{16}}{1} = \frac{3}{16} \text{ miles per minute} \]
In summary, the rate of change of the distance Carol traveled while cross-country skiing is consistently:
\[ \frac{3}{16} \text{ miles per minute} \]
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