To find the interest rate [tex]\( r \)[/tex] for an investment using continuous compounding, we use the formula:
[tex]\[ A = P e^{rt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the future amount,
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the interest rate,
- [tex]\( t \)[/tex] is the time in years,
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
Given:
- [tex]\( P = 2000 \)[/tex] (the initial amount),
- [tex]\( A = 2384.88 \)[/tex] (the amount after 4 years),
- [tex]\( t = 4 \)[/tex] years.
We are tasked with finding the interest rate [tex]\( r \)[/tex].
1. Start with the given formula:
[tex]\[ A = P e^{rt} \][/tex]
2. Substitute the known values:
[tex]\[ 2384.88 = 2000 e^{4r} \][/tex]
3. Solve for [tex]\( e^{4r} \)[/tex]:
[tex]\[ \frac{2384.88}{2000} = e^{4r} \][/tex]
4. Simplify the left side:
[tex]\[ 1.19244 = e^{4r} \][/tex]
5. Take the natural logarithm (ln) of both sides to solve for [tex]\( 4r \)[/tex]:
[tex]\[ \ln(1.19244) = 4r \][/tex]
6. Compute the natural logarithm of 1.19244:
[tex]\[ \ln(1.19244) \approx 0.176 \][/tex]
7. Solve for [tex]\( r \)[/tex] by dividing both sides by 4:
[tex]\[ r = \frac{0.176}{4} \][/tex]
8. Simplify:
[tex]\[ r \approx 0.044 \][/tex]
9. Convert [tex]\( r \)[/tex] to a percentage:
[tex]\[ r \times 100 \approx 4.4\% \][/tex]
So, the interest rate is approximately 4.4%.
