The formula for continuously compounded interest is:
A = Pe^(rt)
where:
A = final amount
P = principal amount
e = Euler's number (approximately 2.71828)
r = annual interest rate (as a decimal)
t = time (in years)
We want to solve for t when A = $2400, P = $465, and r = 0.031. Substituting these values into the formula, we get:
2400 = 465e^(0.031t)
Dividing both sides by 465, we get:
5.16129032 = e^(0.031t)
Taking the natural logarithm of both sides, we get:
ln(5.16129032) = 0.031t
Solving for t, we get:
t = ln(5.16129032)/0.031
t ≈ 22.33 years
Therefore, it will take approximately 22.33 years for Tan's investment to reach $2400.