Question

7. The equation b = 500 (1.05)' gives the balance of a bank account / years since the account was opened. The graph shows the annual account balance for 10 years. . a. What is the average annual rate of change for the bank account? b. Is the average rate of change a good measure of how the bank account varies? Explain your reasoning. account balance (dollars) 1,000 800 600 400 200 1 2 3 4 5 6 7 8 9 10 time since account opening (years)​

Answer 1

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.