To determine how much Ana needs to invest now so that the total amount reaches [tex]$25,000 in 5 years with an interest rate of 4% compounded continuously, we can use the formula for continuously compounded interest:
\[ A = P \cdot e^{(rt)} \]
where:
- \( A \) is the final amount.
- \( P \) is the principal (the initial investment amount).
- \( e \) is the base of the natural logarithms (approximately equal to 2.71828).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( t \) is the time the money is invested for, in years.
We need to solve for \( P \). Rearranging the formula to solve for \( P \):
\[ P = \frac{A}{e^{(rt)}} \]
Given:
- \( A = 25000 \)
- \( r = 0.04 \)
- \( t = 5 \) years
Let's plug in the values:
\[ P = \frac{25000}{e^{(0.04 \times 5)}} \]
First, compute the exponent:
\[ 0.04 \times 5 = 0.20 \]
Next, compute \( e^{0.20} \). Using a calculator, we find that \( e^{0.20} \approx 1.2214 \).
Now, substitute this back into the principal calculation:
\[ P = \frac{25000}{1.2214} \]
Perform the division:
\[ P \approx 20468.27 \]
Therefore, Ana needs to invest approximately $[/tex]20,468.27 now to have a total amount of $25,000 in 5 years, given an interest rate of 4% compounded continuously.
