To determine how many years it will take for an investment to grow to three times its initial value with an interest rate of 5% compounded continuously, we can use the formula for continuous compounding:
[tex]\[ A = P \cdot e^{(rt)} \][/tex]
Where:
- \( A \) is the final amount of the investment,
- \( P \) is the initial amount of the investment,
- \( r \) is the annual interest rate,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given that the final amount \( A \) is three times the initial amount \( P \), we can write the equation as:
[tex]\[ 3P = P \cdot e^{(0.05t)} \][/tex]
To simplify, divide both sides by \( P \) (assuming \( P \neq 0 \)):
[tex]\[ 3 = e^{(0.05t)} \][/tex]
Next, to solve for \( t \), take the natural logarithm (ln) of both sides of the equation:
[tex]\[ \ln(3) = \ln(e^{(0.05t)}) \][/tex]
Using the property of logarithms that \( \ln(e^x) = x \), this simplifies to:
[tex]\[ \ln(3) = 0.05t \][/tex]
Now, isolate \( t \) by dividing both sides by 0.05:
[tex]\[ t = \frac{\ln(3)}{0.05} \][/tex]
Using a calculator to find the natural logarithm of 3:
[tex]\[ \ln(3) \approx 1.0986122886681098 \][/tex]
So:
[tex]\[ t = \frac{1.0986122886681098}{0.05} \approx 21.972245773362193 \][/tex]
Rounding this result to the nearest hundredth, we get:
[tex]\[ t \approx 21.97 \][/tex]
Thus, it will take approximately 21.97 years for the investment to grow to three times its initial value with a continuous compounded annual interest rate of 5%.
