A dance competition on television had five elimination rounds. After each elimination, only one-fourth of the contestants were sent to the next round. The table below shows the number of contestants in each round of the competition:
| x | 1 | 2 | 3 | 4 | 5 |
|---|-----|-----|----|---|---|
| f(x) | 512 | 128 | 32 | 8 | 2 |
Compute the average rate of change of [tex]f(x)[/tex] from [tex]x=2[/tex] to [tex]x=4[/tex], and describe what it represents.
A. -60 contestants per round, and it represents the average rate at which the number of contestants changed from the second round to the fourth round
B. -120 contestants per round, and it represents the average rate at which the number of contestants changed from the second round to the fourth round
C. -60 contestants per round, and it represents the number of contestants who will reach the final round
D. -120 contestants per round, and it represents the number of contestants who will reach the final round
Answer (1)
Calculate the average rate of change using the formula: 4 − 2 f ( 4 ) − f ( 2 ) .
Substitute the values from the table: f ( 2 ) = 128 and f ( 4 ) = 8 .
Compute the average rate of change: 4 − 2 8 − 128 = − 60 .
The average rate of change represents the average decrease in the number of contestants per round from round 2 to round 4. − 60
Explanation
Understanding the Problem We are asked to compute the average rate of change of f ( x ) from x = 2 to x = 4 , and describe what it represents. The average rate of change of a function f ( x ) between two points x 1 and x 2 is given by the formula: x 2 − x 1 f ( x 2 ) − f ( x 1 ) In this case, x 1 = 2 and x 2 = 4 .
Identifying Function Values From the table, we have f ( 2 ) = 128 and f ( 4 ) = 8 .
Calculating Average Rate of Change Now, we can compute the average rate of change: 4 − 2 f ( 4 ) − f ( 2 ) = 4 − 2 8 − 128 = 2 − 120 = − 60 So, the average rate of change is -60.
Interpreting the Result The average rate of change represents the average rate at which the number of contestants changed from the second round to the fourth round. In this case, it means that on average, the number of contestants decreased by 60 contestants per round between the second and fourth rounds.
Examples
Understanding average rates of change is useful in many real-world scenarios. For example, if you are tracking the temperature change over time, the average rate of change can tell you how quickly the temperature is increasing or decreasing. Similarly, in business, you can use the average rate of change to analyze sales trends or customer growth. In this dance competition example, the average rate of change helps us understand how quickly contestants are being eliminated between specific rounds, providing insights into the competition's intensity at different stages.
