To determine how much Charmaine should pay now for a bond that will mature to [tex]$6500 in nine years with an interest rate of 3% per year, compounded continuously, we can use the formula for the present value of a continuously compounded investment:
\[ PV = \frac{FV}{e^{rt}} \]
Where:
- \( PV \) is the present value, or the amount that Charmaine should pay now.
- \( FV \) is the future value, which is $[/tex]6500.
- [tex]\( r \)[/tex] is the annual interest rate, which is 0.03 (3%).
- [tex]\( t \)[/tex] is the time in years, which is 9 years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Let's go through the steps to find the present value:
1. Calculate the exponent:
[tex]\[
rt = 0.03 \times 9 = 0.27
\][/tex]
2. Calculate [tex]\( e^{rt} \)[/tex]:
[tex]\[
e^{0.27} \approx 1.31025
\][/tex]
3. Divide the future value by [tex]\( e^{rt} \)[/tex]:
[tex]\[
PV = \frac{6500}{1.31025} \approx 4961.967
\][/tex]
4. Round the result to the nearest cent:
[tex]\[
PV \approx 4961.97
\][/tex]
So, Charmaine should pay approximately [tex]$4961.97 now for the bond to mature to $[/tex]6500 in nine years, given an interest rate of 3% per year, compounded continuously.
