Final answer:
To find out when Ethan should make the payment, we can use the formula for compound interest and solve for t. By substituting the given values and using logarithms, we find that Ethan should make the payment in approximately 2.29 years.
Step-by-step explanation:
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
- A = Final amount
- P = Initial amount (principle)
- r = Annual interest rate (as a decimal)
- n = Number of times interest is compounded per year
- t = Number of years
In this case, Ethan needs to pay an obligation of $12,000 at an interest rate of 6% compounded monthly. We want to find out when this payment should be given, so we need to solve for t.
Substituting the given values into the formula:
$15,500 = $12,000(1 + 0.06/12)^(12t)
Now we can solve for t by isolating it:
($15,500/$12,000) = (1 + 0.005)^12t
(1.2917) = (1.005)^12t
Using logarithms, we can solve for t:
t = log(1.2917)/log(1.005)
t ≈ 2.2926
Therefore, Ethan should make the payment in approximately 2.29 years.