If 6000 is invested at 5% compounded continuously, what is the amount
after 2 years
The Correct answer and Explanation is:
To calculate the amount of money accumulated after investing $6000 at a continuously compounded interest rate of 5% over 2 years, we use the formula for continuous compounding:A=PertA = Pe^{rt}A=Pert
where:
- AAA is the amount of money accumulated after time ttt,
- PPP is the principal amount (the initial amount of money),
- rrr is the annual interest rate (decimal),
- ttt is the time the money is invested or borrowed for (in years),
- eee is Euler’s number (approximately equal to 2.71828).
Given Values
- P=6000P = 6000P=6000
- r=0.05r = 0.05r=0.05 (5% as a decimal)
- t=2t = 2t=2
Calculation Steps
- Substitute the known values into the formula:
A=6000×e0.05×2A = 6000 \times e^{0.05 \times 2}A=6000×e0.05×2
- Calculate the exponent:
0.05×2=0.10.05 \times 2 = 0.10.05×2=0.1
- Now, substitute 0.10.10.1 back into the equation:
A=6000×e0.1A = 6000 \times e^{0.1}A=6000×e0.1
- Calculate e0.1e^{0.1}e0.1:
Using a calculator, we find that e0.1≈1.10517e^{0.1} \approx 1.10517e0.1≈1.10517.
- Now multiply by the principal:
A≈6000×1.10517≈6630.99A \approx 6000 \times 1.10517 \approx 6630.99A≈6000×1.10517≈6630.99
Result
After 2 years, the amount accumulated from an investment of $6000 at a 5% interest rate compounded continuously is approximately $6630.99.
Explanation
Continuous compounding is an important concept in finance, reflecting how interest can accrue on an investment at every possible moment. Unlike regular compounding, where interest is calculated at specific intervals (annually, semi-annually, etc.), continuous compounding assumes that interest is added instantaneously. The mathematical constant eee is a crucial part of this calculation, representing the limit of (1+1n)n(1 + \frac{1}{n})^n(1+n1)n as nnn approaches infinity, which is foundational in calculus.
In our example, the investment grows from $6000 to about $6630.99 due to the effect of continuous compounding over 2 years. This approach illustrates the power of exponential growth, which significantly increases the total amount as the time period extends or the interest rate rises. Understanding these principles allows investors to make informed decisions about the potential growth of their investments based on different compounding methods.