how much should be invested now at an interest rate of 5% per year, compounded continuously, to have $2000…

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 amount (initial investment), $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We need to solve for $P$, so we can rewrite the formula as $P=\frac{A}{e^{rt}}$.

Step2: Identify the values of $A$, $r$, and $t$

We are given that $A = 2000$, $r=0.05$ (since $5%=0.05$), and $t = 7$.

Step3: Substitute the values into the formula

$P=\frac{2000}{e^{0.05\times7}}=\frac{2000}{e^{0.35}}$.

Step4: Calculate the value of $P$

Using a calculator, $e^{0.35}\approx1.419067$, so $P=\frac{2000}{1.419067}\approx1409.30$.

Answer:

$1409.30$