Answer:
-The balance after 5 years is approximately $7,566.81.
-It will take approximately 11.23 years for the account to earn $1,000 in interest.
Explanation:
To calculate the balance after 5 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the balance after t years, P is the principal, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.
In this case, P = $6,000, r = 0.045 (since the annual interest rate is 4.5%), n = 12 (since interest is compounded monthly), and t = 5. Plugging these values into the formula, we get:
A = $6,000(1 + 0.045/12)^(12*5) ≈ $7,566.81
Therefore, the balance after 5 years is approximately $7,566.81.
To calculate the time it will take for the account to earn $1,000 in interest, we can use the formula:
A = P(1 + r/n)^(nt)
and rearrange it to solve for t:
t = log(A/P) / (n * log(1 + r/n))
where log is the natural logarithm function.
In this case, we want to find t when the interest earned is $1,000, so A = $6,000 + $1,000 = $7,000 (since the interest is added to the principal to get the new balance), P = $6,000, r = 0.045, and n = 12. Plugging these values into the formula, we get:
t = log($7,000/$6,000) / (12 * log(1 + 0.045/12)) ≈ 11.23
Therefore, it will take approximately 11.23 years for the account to earn $1,000 in interest.