if a person earns an annual interest rate of 4% on a savings account, how many years will it take for the…

Explanation:

Step1: Use compound - interest formula

The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the number of years. We want $A = 2P$ and $r=0.04$. Substituting these values into the formula gives $2P=P(1 + 0.04)^t$.

Step2: Simplify the equation

Divide both sides of the equation $2P=P(1 + 0.04)^t$ by $P$ (since $P\neq0$). We get $2=(1.04)^t$.

Step3: Take the natural logarithm of both sides

$\ln(2)=\ln(1.04^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(2)=t\ln(1.04)$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{\ln(1.04)}$. Since $\ln(2)\approx0.6931$ and $\ln(1.04)\approx0.0392$, then $t=\frac{0.6931}{0.0392}\approx17.7$. Rounding up, it takes approximately 18 years.

Answer:

18 years