To determine the amount Nicole should pay for a bond that will mature to [tex]$6000 in six years with an interest rate of 3.5% per year, compounded continuously, we can use the formula for continuous compounding. Let's break it down step-by-step.
### Step-by-Step Solution:
1. Understand the Continuous Compounding Formula:
The formula to calculate the present value \( P \) with continuous compounding is given by:
\[
P = \frac{A}{e^{rt}}
\]
where:
- \( A \) is the maturity amount (the future value, which is $[/tex]6000 in this case),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal, which is 3.5% or 0.035),
- [tex]\( t \)[/tex] is the time in years (which is 6 years),
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
2. Substitute the Given Values into the Formula:
Given:
- [tex]\( A = 6000 \)[/tex]
- [tex]\( r = 0.035 \)[/tex]
- [tex]\( t = 6 \)[/tex]
The formula now looks like this:
[tex]\[
P = \frac{6000}{e^{0.035 \times 6}}
\][/tex]
3. Calculate the Exponent:
First, calculate the exponent [tex]\( rt \)[/tex]:
[tex]\[
rt = 0.035 \times 6 = 0.21
\][/tex]
4. Calculate the Exponential Term:
Next, calculate [tex]\( e^{0.21} \)[/tex]:
[tex]\[
e^{0.21} \approx 1.233678059
\][/tex]
5. Divide the Maturity Amount by the Exponential Term:
Now, divide the maturity amount by the calculated exponential value:
[tex]\[
P = \frac{6000}{1.233678059} \approx 4863.505475821123
\][/tex]
6. Round the Present Value to the Nearest Cent:
Finally, round the calculated present value to the nearest cent:
[tex]\[
P \approx 4863.51
\][/tex]
So, Nicole should pay approximately [tex]$4863.51 for the bond now if it earns interest at a rate of 3.5% per year, compounded continuously, to mature to $[/tex]6000 in six years.
