how long will it take for $1500 to grow to $3000 if it is compounded continuously at a rate of 2.1%?

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 $P = 1500$, $A=3000$, and $r = 0.021$. Substitute these values into the formula: $3000=1500e^{0.021t}$.

Step2: Simplify the equation

Divide both sides of the equation by 1500: $\frac{3000}{1500}=e^{0.021t}$, which simplifies to $2 = e^{0.021t}$.

Step3: Take the natural - logarithm of both sides

Since $\ln(e^{x})=x$, taking the natural logarithm of both sides of the equation $2 = e^{0.021t}$ gives $\ln(2)=\ln(e^{0.021t})$. So, $\ln(2)=0.021t$.

Step4: Solve for $t$

Divide both sides of the equation $\ln(2)=0.021t$ by 0.021: $t=\frac{\ln(2)}{0.021}$. We know that $\ln(2)\approx0.693147$, so $t=\frac{0.693147}{0.021}\approx33$.

Answer:

Approximately 33 years.