To determine the amount that the investment is worth at the end of the given time period with continuous compounding, we use the formula for continuous compounding:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [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.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- [tex]\( P = 8000 \)[/tex] dollars
- [tex]\( t = 11 \)[/tex] years
Let's calculate for each interest rate separately.
### (a) 2% Interest
1. Convert the interest rate to a decimal:
[tex]\[ r = 0.02 \][/tex]
2. Substitute the values into the formula:
[tex]\[ A = 8000 \cdot e^{0.02 \cdot 11} \][/tex]
3. Calculate the exponent:
[tex]\[ 0.02 \times 11 = 0.22 \][/tex]
4. Use the value of [tex]\( e^{0.22} \approx 1.246 \)[/tex]:
[tex]\[ A = 8000 \cdot 1.246 \][/tex]
5. Calculate the final amount:
[tex]\[ A = 9968.61 \][/tex]
Therefore, the amount after 11 years at a 2% interest rate is [tex]\(\$ 9968.61\)[/tex].
### (b) 3% Interest
1. Convert the interest rate to a decimal:
[tex]\[ r = 0.03 \][/tex]
2. Substitute the values into the formula:
[tex]\[ A = 8000 \cdot e^{0.03 \cdot 11} \][/tex]
3. Calculate the exponent:
[tex]\[ 0.03 \times 11 = 0.33 \][/tex]
4. Use the value of [tex]\( e^{0.33} \approx 1.391 \)[/tex]:
[tex]\[ A = 8000 \cdot 1.391 \][/tex]
5. Calculate the final amount:
[tex]\[ A = 11127.75 \][/tex]
Therefore, the amount after 11 years at a 3% interest rate is [tex]\(\$ 11127.75\)[/tex].
### (c) 6.5% Interest
1. Convert the interest rate to a decimal:
[tex]\[ r = 0.065 \][/tex]
2. Substitute the values into the formula:
[tex]\[ A = 8000 \cdot e^{0.065 \cdot 11} \][/tex]
3. Calculate the exponent:
[tex]\[ 0.065 \times 11 = 0.715 \][/tex]
4. Use the value of [tex]\( e^{0.715} \approx 2.044 \)[/tex]:
[tex]\[ A = 8000 \cdot 2.044 \][/tex]
5. Calculate the final amount:
[tex]\[ A = 16353.49 \][/tex]
Therefore, the amount after 11 years at a 6.5% interest rate is [tex]\(\$ 16353.49\)[/tex].
