Let’s solve the problem step-by-step to determine how many years are required for an investment to double in value if it appreciates at a rate of 3% compounded continuously.
1. Understand the problem:
- We have an initial investment that is continuously compounded at an annual interest rate of 3%.
- We want to find the time required for this investment to double in value.
2. Use the formula for continuous compounding:
The formula for the future value [tex]\( A \)[/tex] of an investment when compounded continuously is given by:
[tex]\[
A = P \cdot e^{rt}
\][/tex]
Here,
- [tex]\( P \)[/tex] is the initial principal (the starting amount of the investment).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
3. Apply the given conditions:
- The initial value [tex]\( P \)[/tex] can be any positive number. For simplicity, we can consider it as [tex]\( 1 \)[/tex] (since any positive number will double under the same conditions).
- The final value [tex]\( A \)[/tex] after doubling would be [tex]\( 2P \)[/tex], which is [tex]\( 2 \times 1 = 2 \)[/tex].
- The interest rate [tex]\( r \)[/tex] is 3%, which is [tex]\( 0.03 \)[/tex] when expressed as a decimal.
4. Set up the equation:
[tex]\[
2 = 1 \cdot e^{0.03t}
\][/tex]
Simplifying this, we get:
[tex]\[
2 = e^{0.03t}
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
To isolate [tex]\( t \)[/tex], take the natural logarithm (ln) of both sides:
[tex]\[
\ln(2) = 0.03t
\][/tex]
Now solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(2)}{0.03}
\][/tex]
6. Calculate the value:
Using a calculator, we find that:
[tex]\[
\ln(2) \approx 0.693147
\][/tex]
Therefore:
[tex]\[
t = \frac{0.693147}{0.03} \approx 23.1049
\][/tex]
7. Round the answer to one decimal place:
[tex]\[
t \approx 23.1
\][/tex]
Thus, the number of years required for an investment to double in value with a continuous compounding interest rate of 3% is approximately 23.1 years.
