To solve this problem, we need to find the present value (initial amount) of a bond that matures to [tex]$4500 in eight years with an interest rate of 3.5% per year, compounded continuously. Let's go through the calculations step-by-step.
### Step-by-Step Solution:
1. Identify the given values:
- Future Value (FV) = $[/tex]4500
- Annual Interest Rate (r) = 3.5%
- Time (t) = 8 years
2. Convert the annual interest rate percentage to a decimal:
- [tex]\( r = \frac{3.5}{100} = 0.035 \)[/tex]
3. Use the continuous compounding formula to find the present value (PV):
The formula for continuous compounding is:
[tex]\[
A = P \cdot e^{rt}
\][/tex]
Here, [tex]\( A \)[/tex] is the future value, [tex]\( P \)[/tex] is the present value (initial amount), [tex]\( e \)[/tex] is the base of the natural logarithm, [tex]\( r \)[/tex] is the annual interest rate, and [tex]\( t \)[/tex] is the time in years.
We need to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{A}{e^{rt}}
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
P = \frac{4500}{e^{0.035 \cdot 8}}
\][/tex]
5. Calculate the exponent:
[tex]\[
0.035 \cdot 8 = 0.28
\][/tex]
6. Calculate [tex]\( e^{0.28} \)[/tex]:
After calculating [tex]\( e^{0.28} \)[/tex], we get approximately 1.32313.
7. Divide the future value by [tex]\( e^{0.28} \)[/tex]:
[tex]\[
P = \frac{4500}{1.32313}
\][/tex]
8. Perform the division to find the present value:
[tex]\[
P \approx 3401.03
\][/tex]
### Final Answer:
Dan should pay $3401.03 for the bond now.
