Final answer:
To calculate Amelia's annual interest rate, the compound interest formula is used with her starting and ending balance over a period of four years. Dividing the final amount by the principal amount and then taking the fourth root before subtracting 1 yields the rate at which interest was earned.
Step-by-step explanation:
To find the rate at which Amelia earned interest on her bank account, we can use the compound interest formula:
Final Amount = Principal \(\times\) \((1 + rate)^{time}\)
Given Amelia's initial balance of $200 (the principal) and her balance at the end of four years being $220.82, we can set up the equation:
$220.82 = $200 \(\times\) \((1 + r)^4\)
Divide both sides by $200:
1.1041 = \((1 + r)^4\)
To find the interest rate, we take the fourth root of 1.1041:
1 + r = \(\sqrt[4]{1.1041}\)
Subtract 1 from both sides to get r:
r = \(\sqrt[4]{1.1041}\) - 1
Calculating the fourth root of 1.1041 and subtracting 1 gives us Amelia's annual interest rate.