Ethan deposits dollar sign, 480$480 every month into an account earning an annual interest rate of 4.5%, compounded monthly. How many years would it be until Ethan had dollar sign, 48, comma, 000$48,000 in the account, to the nearest tenth of a year? Use the following formula to determine your answer.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years the money is invested or borrowed for
In this case:
P = $480
r = 4.5% = 0.045
n = 12 (monthly compounding)
A = $48,000
We want to find t, the number of years.
$48,000 = $480(1 + 0.045/12)^(12t)
$100 = (1 + 0.00375)^(12t)
$100 = (1.00375)^(12t)
Now, we need to solve for t by taking the natural logarithm of both sides:
ln($100) = ln((1.00375)^(12t))
ln($100) = 12t * ln(1.00375)
ln($100) = 12t * 0.003744994
ln($100) = 0.04493993t
Now, solve for t:
t = ln($100) / 0.04493993
t ≈ 29 years
Therefore, it would take approximately 29 years for Ethan to have $48,000 in the account.