To find how long it takes for an investment to double when it is compounded semi-annually at an annual interest rate of 8%, we can use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the principal (initial amount)
- [tex]\( r \)[/tex] is the annual interest rate (decimal)
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year
- [tex]\( t \)[/tex] is the number of years the money is invested
Given:
- The principal [tex]\( P \)[/tex] is [tex]$1,300.00
- The final amount \( A \) we want is $[/tex]2,600.00 (double the principal)
- The annual interest rate [tex]\( r \)[/tex] is 8%, which is 0.08 in decimal form
- The interest is compounded semi-annually, so [tex]\( n = 2 \)[/tex]
The formula becomes:
[tex]\[ 2,600 = 1,300 \left(1 + \frac{0.08}{2} \right)^{2t} \][/tex]
First, simplify the equation:
[tex]\[ 2,600 = 1,300 \left(1 + 0.04 \right)^{2t} \][/tex]
[tex]\[ 2,600 = 1,300 \left(1.04 \right)^{2t} \][/tex]
Next, divide both sides by 1,300 to isolate the exponential term:
[tex]\[ 2 = \left(1.04 \right)^{2t} \][/tex]
To solve for [tex]\( t \)[/tex], take the natural logarithm (ln) of both sides:
[tex]\[ \ln(2) = \ln\left(\left(1.04 \right)^{2t}\right) \][/tex]
Using the property of logarithms [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex], rewrite the equation:
[tex]\[ \ln(2) = 2t \cdot \ln(1.04) \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(2)}{2 \cdot \ln(1.04)} \][/tex]
Evaluating the logarithms and dividing:
[tex]\[ t \approx \frac{0.693}{2 \cdot 0.039} \][/tex]
[tex]\[ t \approx \frac{0.693}{0.078} \][/tex]
[tex]\[ t \approx 8.836 \][/tex]
Therefore, it will take approximately [tex]\( 8.836 \)[/tex] years for the investment to double when compounded semi-annually at an 8% annual interest rate.
