To solve this problem using the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n} \right)^{nt}
\][/tex]
We need to find the time [tex]\( t \)[/tex] it takes for the initial amount [tex]\( P \)[/tex] to double given the annual interest rate [tex]\( r \)[/tex] compounded quarterly.
1. Identify given values:
- Initial principal [tex]\( P = \$1200.00 \)[/tex]
- Final amount [tex]\( A \)[/tex] when the investment doubles, so [tex]\( A = 2 \times P = 2 \times 1200.00 = \$2400.00 \)[/tex]
- Annual interest rate [tex]\( r = 0.03 \)[/tex]
- Number of times interest is compounded per year [tex]\( n = 4 \)[/tex]
2. Set up the compound interest formula:
[tex]\[
2400 = 1200 \left(1 + \frac{0.03}{4} \right)^{4t}
\][/tex]
3. Simplify inside the parentheses:
[tex]\[
1 + \frac{0.03}{4} = 1 + 0.0075 = 1.0075
\][/tex]
4. Substitute and simplify the equation:
[tex]\[
2400 = 1200 \left(1.0075\right)^{4t}
\][/tex]
5. Divide both sides of the equation by 1200:
[tex]\[
2 = \left(1.0075\right)^{4t}
\][/tex]
6. Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln(2) = \ln\left(\left(1.0075\right)^{4t}\right)
\][/tex]
7. Apply the power rule of logarithms [tex]\( \ln\left(a^b\right) = b \ln(a) \)[/tex]:
[tex]\[
\ln(2) = 4t \cdot \ln(1.0075)
\][/tex]
8. Isolate [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(2)}{4 \cdot \ln(1.0075)}
\][/tex]
9. Calculate the values using known logarithms:
[tex]\[
t = \frac{\ln(2)}{4 \cdot \ln(1.0075)} \approx \frac{0.693147}{4 \cdot 0.007481} \approx \frac{0.693147}{0.029924}
\][/tex]
10. Complete the division to find [tex]\( t \)[/tex]:
[tex]\[
t \approx 23.191
\][/tex]
So, it will take approximately 23.191 years for the initial amount of [tex]\( \$1200.00 \)[/tex] to double when invested at an interest rate of [tex]\( 3\% \)[/tex] compounded quarterly.
