18. if you deposit $1,000 into an account that pays 4% interest compounded continuously, how long will it…

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, $r$ is the annual interest rate (in decimal form), and $t$ is the time in years. We are given that $P = 1000$, $A = 2000$, and $r=0.04$. Substitute these values into the formula: $2000 = 1000e^{0.04t}$

Step2: Simplify the equation

Divide both sides of the equation by 1000: $\frac{2000}{1000}=e^{0.04t}$, which simplifies to $2 = e^{0.04t}$

Step3: Take the natural logarithm of both sides

Since $\ln(e^{x})=x$, taking the natural logarithm of both sides of the equation $2 = e^{0.04t}$ gives: $\ln(2)=\ln(e^{0.04t})$, so $\ln(2) = 0.04t$

Step4: Solve for $t$

Divide both sides of the equation by 0.04: $t=\frac{\ln(2)}{0.04}$ We know that $\ln(2)\approx0.6931$, so $t=\frac{0.6931}{0.04}=17.3275\approx17.33$ years

Answer:

$t\approx17.33$ years