You began membership in a new health and fitness club which has access to a dietician and personal trainer. They help you develop a special eight-week diet and exercise program. The data in the following table represents your weight, [tex]$w$[/tex], as a function of time, [tex]$t$[/tex], over an eight-week period.
| Time (weeks) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| :----------- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- |
| Weight (lb) | 150 | 144 | 143 | 141 | 140 | 137 | 137 | 132 | 129 |
During which time period is the average rate of change zero?
A. From week 7 to week 8
B. From week 3 to week 4
C. From week 0 to week 1
D. From week 5 to week 6
Answer (2)
Calculate the average rate of change for each given time period using the formula Δ t Δ w .
From week 7 to week 8: The average rate of change is -3.
From week 3 to week 4: The average rate of change is -1.
From week 0 to week 1: The average rate of change is -6.
From week 5 to week 6: The average rate of change is 0, so the final answer is d .
Explanation
Understanding the Problem We are given a table that shows your weight ( w ) over an eight-week period ( t ). The question asks us to find the time period during which the average rate of change of your weight is zero. The average rate of change is calculated as the change in weight divided by the change in time, or Δ t Δ w .
Plan of Action To find the time period with an average rate of change of zero, we need to calculate the average rate of change for each of the given options and see which one equals zero.
Calculating Average Rates of Change Let's calculate the average rate of change for each option:
a. From week 7 to week 8: 8 − 7 129 − 132 = 1 − 3 = − 3
b. From week 3 to week 4: 4 − 3 140 − 141 = 1 − 1 = − 1
c. From week 0 to week 1: 1 − 0 144 − 150 = 1 − 6 = − 6
d. From week 5 to week 6: 6 − 5 137 − 137 = 1 0 = 0
Identifying the Zero Rate of Change From the calculations above, we can see that the average rate of change is zero during the time period from week 5 to week 6.
Final Answer Therefore, the correct answer is d. From week 5 to week 6 .
Examples
Understanding rates of change is crucial in many real-world scenarios. For instance, in finance, it helps in analyzing investment growth or decline over time. In environmental science, it's used to track changes in pollution levels or species populations. In our case, it helps to understand weight loss progress. By calculating the rate of change, you can see how quickly you are losing weight and adjust your diet and exercise plan accordingly. This concept applies not only to weight management but also to tracking progress in various aspects of life, such as learning a new skill or achieving a fitness goal.
The average rate of change of weight is zero from week 5 to week 6, as the weight remains the same during this period. Therefore, the answer is option D. This indicates that no weight was lost between week 5 and week 6.
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