To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the amount of money after t years
P = the principal amount (the initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
We know that P = $5000, A = $6000, r = 0.09 (9%), and n = 12 (since interest is compounded monthly). We can solve for t algebraically:
A = P(1 + r/n)^(nt)
$6000 = $5000(1 + 0.09/12)^(12t)
$6000/$5000 = (1 + 0.09/12)^(12t)
1.2 = (1.0075)^(12t)
log(1.2) = log(1.0075)^(12t)
log(1.2) = 12t * log(1.0075)
t = log(1.2) / (12 * log(1.0075))
t ≈ 4.58
Therefore, it will take about 4.58 years (or about 55 months) for $5000 to grow to $6000 at an annual rate of 9% compounded monthly.