Answer:
The formula for continuous compounding is given as:
A = Pe^(rt)
where A is the final amount, P is the initial amount, r is the annual interest rate (as a decimal), t is the time in years, and e is the mathematical constant approximately equal to 2.71828.
To find the time for an investment to double, we can set A/P = 2 and solve for t:
2 = e^(rt)
Taking the natural logarithm of both sides, we get:
ln(2) = rt
Solving for t, we get:
t = ln(2)/r
Substituting r = 0.0575 (since 5.75% = 0.0575 as a decimal), we get:
t = ln(2)/0.0575 ≈ 12.06 years
Therefore, at an annual interest rate of 5.75% compounded continuously, it takes approximately 12.06 years for an investment to double.