Final answer:
The money was invested for approximately 5 years and 7 months.
Step-by-step explanation:
To determine the number of years and months for the investment, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount (including the interest)
- P is the principal (initial amount)
- r is the annual interest rate (expressed as a decimal)
- n is the number of times interest is compounded per year
- t is the number of years
In this case, we have A = $1149 + $417 = $1566, P = $1149, r = 0.1, and n = 12 (compounded monthly). We need to solve for t:
$1566 = $1149(1 + 0.1/12)^(12t)
Dividing both sides by $1149:
1.3627 = (1.0083)^(12t)
Taking the natural logarithm of both sides:
ln(1.3627) = ln((1.0083)^(12t))
Using the property of logarithms, we can bring down the exponent:
12t ln(1.0083) = ln(1.3627)
Dividing both sides by 12 ln(1.0083):
t = ln(1.3627) / (12 ln(1.0083))
Using a calculator, we find t ≈ 5.59 years.
Since the interest is compounded monthly, we can convert the remaining decimals into months:
0.59 years ≈ 0.59 * 12 ≈ 7.08 months.
Therefore, the money was invested for approximately 5 years and 7 months.