To determine the amount Donna should pay for the bond now, we need to find the present value of the bond, which matures to [tex]$6000 in nine years at an interest rate of 3% per year, compounded continuously.
The formula for continuous compounding is given by:
\[ PV = MV \times e^{-rt} \]
where:
- \( PV \) is the Present Value (the amount Donna should pay now),
- \( MV \) is the Maturity Value (the amount the bond will be worth in the future, $[/tex]6000 in this case),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal, so 3% becomes 0.03),
- [tex]\( t \)[/tex] is the time in years (9 years in this case),
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
Now we can plug in the values and solve for the Present Value [tex]\( PV \)[/tex]:
[tex]\[ PV = 6000 \times e^{-0.03 \times 9} \][/tex]
First, calculate the exponent:
[tex]\[ -0.03 \times 9 = -0.27 \][/tex]
Then, raise [tex]\( e \)[/tex] to the power of [tex]\(-0.27\)[/tex]:
[tex]\[ e^{-0.27} \approx 0.763379494 \][/tex]
Now, multiply this result by the Maturity Value ([tex]$6000):
\[ PV = 6000 \times 0.763379494 \approx 4580.276966021119 \]
Finally, to round the result to the nearest cent:
\[ PV \approx 4580.28 \]
Therefore, Donna should pay approximately $[/tex]4580.28 for the bond now, given that it matures to $6000 in nine years with an interest rate of 3% compounded continuously.
