how much should be invested now at an interest rate of 5.5% per year, compounded continuously, to have $2000…

Explanation:

Step1: Recall the 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 are given that $A=$2000$, $r = 0.055$ (since $5.5%=0.055$), and $t = 4$. We need to solve for $P$. From $A = Pe^{rt}$, we can isolate $P$ by the formula $P=\frac{A}{e^{rt}}$.

Step2: Substitute the given values into the formula

Substitute $A = 2000$, $r=0.055$, and $t = 4$ into $P=\frac{A}{e^{rt}}$. We get $P=\frac{2000}{e^{(0.055\times4)}}=\frac{2000}{e^{0.22}}$. Since $e^{0.22}\approx1.246077$, then $P=\frac{2000}{1.246077}$.

Step3: Calculate the value of $P$

$P=\frac{2000}{1.246077}\approx1604.9$

Answer:

$$1604.90$