To find how much money needs to be invested initially to achieve a future amount of [tex]$34,000 after 18 years at an interest rate of 2.9% compounded continuously, we can use the continuous compound interest formula:
\[ A = P \cdot e^{rt} \]
Where:
- \(A\) is the future amount of money or desired amount (\$[/tex]34,000).
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money we're trying to find).
- [tex]\(r\)[/tex] is the annual interest rate (2.9% or 0.029 as a decimal).
- [tex]\(t\)[/tex] is the time the money is invested for in years (18 years).
- [tex]\(e\)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
We need to find [tex]\(P\)[/tex]. We start by rearranging the formula to solve for [tex]\(P\)[/tex]:
[tex]\[ P = \frac{A}{e^{rt}} \][/tex]
Now, substitute the given values into the formula:
[tex]\[ P = \frac{34,000}{e^{0.029 \cdot 18}} \][/tex]
Calculating the exponent:
[tex]\[ 0.029 \times 18 = 0.522 \][/tex]
Then, [tex]\( e^{0.522} \approx 1.68555 \)[/tex].
Now, we can plug this value into the formula to find [tex]\( P \)[/tex]:
[tex]\[ P = \frac{34,000}{1.68555} \][/tex]
[tex]\[ P \approx 20,173.31 \][/tex]
Therefore, the principal amount that needs to be invested is approximately [tex]$20,173.31.
Thus, the correct answer is:
A. $[/tex]20,173.31
