suppose that $5000 is placed in a savings account at an annual rate of 9.4%, compounded quarterly. assuming…

Explanation:

Step1: Identify compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Given $P = 5000$, $r=0.094$ (since $9.4%=0.094$), $n = 4$ (compounded quarterly), and $A = 8935$. Substitute these values into the formula: $8935=5000(1 +\frac{0.094}{4})^{4t}$.

Step2: Simplify the equation

First, divide both sides of the equation by $5000$: $\frac{8935}{5000}=(1 + 0.0235)^{4t}$. $1.787=(1.0235)^{4t}$.

Step3: Take the natural logarithm of both sides

$\ln(1.787)=\ln((1.0235)^{4t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get: $\ln(1.787)=4t\ln(1.0235)$.

Step4: Solve for $t$

First, find $\ln(1.787)\approx0.570$ and $\ln(1.0235)\approx0.0232$. Then, $t=\frac{\ln(1.787)}{4\ln(1.0235)}$. $4\ln(1.0235)=4\times0.0232 = 0.0928$. $t=\frac{0.570}{0.0928}\approx6.14$.

Answer:

$6.14$