Final answer:
The average annual rate of change for the bank account is $31.45 per year. However, this rate is not a good measure of how the bank account actually varies because it doesn't account for the compounding effect, which means the account's growth accelerates over time.
Step-by-step explanation:
Calculating Average Annual Rate of Change
The equation b = 500(1.05)^t describes the growth of a bank account balance over time, where b represents the balance, and t represents time in years.
To find the average annual rate of change, we consider the balance at the start (t = 0), which is $500, and the balance at the end of 10 years (t = 10).
Using the given formula, the balance after 10 years is 500(1.05)^10. Calculating this, we get a balance of $814.45. The average annual rate of change is then the difference in balance divided by the number of years:
Average Rate of Change = (Final Balance - Initial Balance) / Number of Years
In this case:
Average Rate of Change = (814.45 - 500) / 10
= 314.45 / 10
= $31.45 per year.
Assessing the Measure of Variability
The average rate of change is a linear measure.
However, since the account earns interest compounded annually, the growth is exponential, and the balance increases by a greater amount each subsequent year.
Therefore, the average rate of change does not accurately represent how the bank account varies over time; it merely provides a simplified view of the overall increase from start to finish. It doesn't reflect the increasing growth rate due to compounding.