Question 1: Jumping Jacks
Let's analyze the data for Kimberly and Katrina.
a. Which person is doing more jumping jacks per second?
To find out how many jumping jacks each person is doing per second, we calculate the rate for each time interval:
Kimberly:
- From 0 to 3 seconds: 17 jumping jacks / 3 seconds = 5.67 jacks/second
- From 3 to 8 seconds: (37 - 17) / (8 - 3) = 20 / 5 = 4.0 jacks/second
- From 8 to 12 seconds: (53 - 37) / (12 - 8) = 16 / 4 = 4.0 jacks/second
- From 12 to 16 seconds: (69 - 53) / (16 - 12) = 16 / 4 = 4.0 jacks/second
Average rate for Kimberly: \((5.67 + 4.0 + 4.0 + 4.0) / 4 = 4.67\) jacks/second.
Katrina:
- From 0 to 2 seconds: 10 jumping jacks / 2 seconds = 5.0 jacks/second
- From 2 to 5 seconds: (25 - 10) / (5 - 2) = 15 / 3 = 5.0 jacks/second
- From 5 to 12 seconds: (60 - 25) / (12 - 5) = 35 / 7 = 5.0 jacks/second
- From 12 to 20 seconds: (100 - 60) / (20 - 12) = 40 / 8 = 5.0 jacks/second
Average rate for Katrina: \(5.0\) jacks/second.
Conclusion: Kimberly's average rate is 4.67 jacks/second, and Katrina's average rate is 5.0 jacks/second. Thus, Katrina is doing more jumping jacks per second.
b. Which person had done more jumping jacks initially before the timer started?
- Kimberly's initial count: 0 jumping jacks (since the data starts from 3 seconds).
- Katrina's initial count: 0 jumping jacks (since the data starts from 2 seconds).
Conclusion: Both had done 0 jumping jacks initially before their respective timers started.
c. Which person shows a proportional relationship?
A proportional relationship means that the number of jumping jacks is directly proportional to the time. This can be identified if the ratio of jumping jacks to time remains constant.
For Kimberly: At intervals, the ratios are different, it does not have the same rate.
For Katrina: The rates are constant at 5.0 jacks/second.
Conclusion: Katrina shows a proportional relationship.
Question 3: Functions
We have the following functions:
- Function A: \(y = mx + b\) (but we need more details to analyze)
- Function B: \(y = mx + b\) (again, more details required)
- Function C: \(y = x + 5\)
Let's assume Function A and Function B have a given slope \(m\) and intercept \(b\) but were not specified. As we only have Function C explicitly, we can analyze it.
a. Which function has the greatest rate of change (slope)?
If we can only analyze Function C, the slope is 1 (from \(y = x + 5\)). Without values for Function A and B, we cannot determine their slopes. If other functions have designated slopes, compare them to find the highest slope.
b. Which function has the smallest y-intercept?
For Function C, the \(y\)-intercept is \(5\). We need the intercepts of Functions A and B to compare. The one with the smallest \(b\) would have the smallest \(y\)-intercept.
c. Which function would you want to represent your money earned per hour?
You would typically want a function with the highest positive slope since that would represent a greater increase in earnings over time. Without A and B's slopes, we can't compare definitively.
d. Which function would give you the highest initial amount of money?
This depends on the \(y\)-intercept. The function with the highest \(b\) value would represent the highest initial amount. Again, we only know that Function C has a \(y\)-intercept of \(5\).
Conclusion for this question: More information about Functions A and B is needed to provide definitive answers. Adjustments in rates and intercepts can greatly change the conclusions.
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