Final answer:
To find the interest rate of the second account that yields the same interest earned as the first account, we use the formula for continuous compounding and solve for the second account's interest rate. Assuming a principal of $1000, the interest rate of the second account should be approximately 3.81%.
Step-by-step explanation:
To find the interest rate of the second account, we need to determine the rate that will yield the same interest earned over one year as the first account. Let's call the interest rate of the second account 'x'.
The formula to calculate the interest earned over one year with continuous compounding is:
A = P * e^(rt),
where A is the final amount, P is the principal, r is the interest rate, and t is the time in years.
For the first account, the interest rate is 3.78% APR, which can be converted to a decimal as r = 0.0378. The principal (P) is not given in the question, so we can assume any amount for the sake of comparison.
Let's assume P = $1000. Plugging in the values, we get
A = 1000 * e^(0.0378 * 1) = 1000 * e^0.0378 = 1000 * 1.038676776 = $1038.68 (approximately).
Now, to find the interest rate of the second account, we need to make the interest earned over one year the same as the first account. So we set up the equation:
$1000 * e^(x * 1) = $1038.68.
Solving for x, we divide both sides by $1000 and take the natural logarithm of both sides:
x = ln(1038.68/1000) = ln(1.03868) = 0.038097364 (approximately).
Therefore, the interest rate of the second account should be approximately 3.81% in order to earn the same interest as the first account.