To determine how long it will take for the investment to grow to \[tex]$44,000, we need to solve for \( t \) in the given equation:
\[
S = 11,000 \left(1 + \frac{0.12}{12}\right)^{12t}
\]
Given that \( S = 44,000 \), we can set up the equation as follows:
\[
44,000 = 11,000 \left(1 + \frac{0.12}{12}\right)^{12t}
\]
First, simplify the expression inside the parentheses:
\[
1 + \frac{0.12}{12} = 1 + 0.01 = 1.01
\]
So the equation becomes:
\[
44,000 = 11,000 \times (1.01)^{12t}
\]
Next, divide both sides of the equation by 11,000 to isolate the exponential term:
\[
\frac{44,000}{11,000} = (1.01)^{12t}
\]
\[
4 = (1.01)^{12t}
\]
To solve for \( t \), take the natural logarithm of both sides:
\[
\ln(4) = \ln((1.01)^{12t})
\]
Using the power rule of logarithms \( \ln(a^b) = b \ln(a) \), we can rewrite the equation as:
\[
\ln(4) = 12t \ln(1.01)
\]
Now, solve for \( t \) by dividing both sides by \( 12 \ln(1.01) \):
\[
t = \frac{\ln(4)}{12 \ln(1.01)}
\]
Calculate the natural logarithms:
\[
\ln(4) \approx 1.3863
\]
\[
\ln(1.01) \approx 0.00995
\]
Substitute these values into the equation:
\[
t = \frac{1.3863}{12 \times 0.00995}
\]
Multiply the denominator:
\[
12 \times 0.00995 \approx 0.1194
\]
Finally, divide to find \( t \):
\[
t = \frac{1.3863}{0.1194} \approx 11.61
\]
Therefore, it will take approximately \( 11.61 \) years for the investment to grow to \$[/tex]44,000.
The investment will grow to \$44,000 in [tex]\( 11.61 \)[/tex] years.
