To determine how much should be invested now at an interest rate of 6.5% per year, compounded continuously, to have [tex]$2500 in four years, we use the formula for continuous compounding.
The formula for the future value \(FV\) with continuous compounding is:
\[ FV = PV \cdot e^{(r \cdot t)} \]
where:
- \(FV\) is the future value,
- \(PV\) is the present value (the amount to be invested),
- \(r\) is the annual interest rate (expressed as a decimal),
- \(t\) is the time the money is invested for in years,
- \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
We can rearrange this formula to solve for the present value \(PV\):
\[ PV = \frac{FV}{e^{(r \cdot t)}} \]
Given:
- \(FV = \$[/tex]2500\),
- [tex]\(r = 0.065\)[/tex] (which is 6.5% expressed as a decimal),
- [tex]\(t = 4\)[/tex] years,
We will calculate the present value [tex]\(PV\)[/tex] by substituting these values into the formula.
Step-by-step:
1. First, calculate the exponent [tex]\(r \cdot t\)[/tex]:
[tex]\[ r \cdot t = 0.065 \times 4 = 0.26 \][/tex]
2. Then, calculate [tex]\(e^{0.26}\)[/tex].
3. Next, divide the future value [tex]\(FV\)[/tex] by [tex]\(e^{0.26}\)[/tex]:
[tex]\[ PV = \frac{2500}{e^{0.26}} \][/tex]
Upon calculation, the present value [tex]\(PV\)[/tex] is found to be approximately 1927.63 when rounded to the nearest cent.
Thus, the amount that should be invested now to have [tex]$2500 in four years, compounded continuously at an interest rate of 6.5% per year, is \$[/tex]1927.63.
