21. how much should be invested now to have $5000 in 12 years at 5.5% interest compounded continuously?

Explanation:

Step1: Recall continuous - compounding formula

The formula for continuous - compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We want to find $P$, and we can re - arrange the formula to $P=\frac{A}{e^{rt}}$.

Step2: Convert the interest rate to decimal

The annual interest rate $r = 5.5%=0.055$, the time $t = 12$ years, and the final amount $A = 5000$.

Step3: Substitute values into the formula

$P=\frac{5000}{e^{0.055\times12}}$. First, calculate the exponent: $0.055\times12 = 0.66$. Then, find $e^{0.66}\approx1.9348$. So, $P=\frac{5000}{1.9348}\approx2584.27$.

Answer:

$$2584.27$