Carol is cross-country skiing. The table shows the distance she traveled after various numbers of minutes. What is the rate of change?
| Minutes | Distance Traveled (miles) |
|---|---|
| 2 | [$\frac{1}{6}$] |
| 3 | [$\frac{17}{48}$] |
| 4 | [$\frac{13}{24}$] |
| 5 | [$\frac{35}{48}$] |
| 6 | [$\frac{11}{12}$]
Answer (2)
Calculate the rate of change between the first two points: 3 − 2 48 17 − 6 1 = 16 3 .
Calculate the rate of change between the second and third points: 4 − 3 24 13 − 48 17 = 16 3 .
Calculate the rate of change between the third and fourth points: 5 − 4 48 35 − 24 13 = 16 3 .
Calculate the rate of change between the fourth and fifth points: 6 − 5 12 11 − 48 35 = 16 3 .
The rate of change is constant, so the rate of change is 16 3 .
Explanation
Understanding the Problem We are given a table showing the distance Carol traveled while cross-country skiing over a period of minutes. We want to find the rate of change of the distance with respect to time. The rate of change can be found by calculating the change in distance divided by the change in time between any two points in the table.
Calculating Rate of Change Let's calculate the rate of change between the first two data points. The first point is (2, 6 1 ) and the second point is (3, 48 17 ). The change in distance is 48 17 − 6 1 = 48 17 − 48 8 = 48 9 = 16 3 . The change in time is 3 − 2 = 1 . Therefore, the rate of change is 16 3 /1 = 16 3 .
Verifying Constant Rate of Change Now, let's calculate the rate of change between the second and third data points. The second point is (3, 48 17 ) and the third point is (4, 24 13 ). The change in distance is 24 13 − 48 17 = 48 26 − 48 17 = 48 9 = 16 3 . The change in time is 4 − 3 = 1 . Therefore, the rate of change is 16 3 /1 = 16 3 .
Confirming Constant Rate We can continue this process for the remaining data points. The rate of change between the third and fourth points is 48 35 − 24 13 = 48 35 − 48 26 = 48 9 = 16 3 , and the change in time is 5 − 4 = 1 , so the rate of change is 16 3 . The rate of change between the fourth and fifth points is 12 11 − 48 35 = 48 44 − 48 35 = 48 9 = 16 3 , and the change in time is 6 − 5 = 1 , so the rate of change is 16 3 .
Final Answer Since the rate of change is constant between all consecutive points, the rate of change is 16 3 miles per minute.
Examples
Understanding rate of change is crucial in many real-world scenarios. For instance, if you're tracking the speed of a car, the rate of change tells you how much the car's speed is increasing or decreasing per unit of time. Similarly, in financial analysis, the rate of change can represent the growth rate of an investment or the rate of inflation. In this problem, we found that Carol's rate of change while cross-country skiing is constant, meaning she covers the same distance every minute. This concept helps in predicting future outcomes based on current trends.
The rate of change in distance traveled by Carol while cross-country skiing is 16 3 miles per minute, which is constant across all time intervals in the data provided.
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