suppose that $5000 is placed in a savings account at an annual rate of 6.4%, compounded quarterly. assuming…

Explanation:

Step1: Recall 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 $P = 5000$, $A=7675$, $r = 0.064$ (since $6.4%=0.064$), and $n = 4$ (compounded quarterly). Substitute these values into the formula: $7675=5000(1 +\frac{0.064}{4})^{4t}$.

Step2: Simplify the equation

First, simplify $1+\frac{0.064}{4}=1 + 0.016=1.016$. The equation becomes $\frac{7675}{5000}=(1.016)^{4t}$. $\frac{7675}{5000}=1.535$, so $1.535=(1.016)^{4t}$.

Step3: Take the natural logarithm of both sides

$\ln(1.535)=\ln((1.016)^{4t})$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(1.535)=4t\ln(1.016)$.

Step4: Solve for $t$

We know that $\ln(1.535)\approx0.428$ and $\ln(1.016)\approx0.0159$. So, $t=\frac{\ln(1.535)}{4\ln(1.016)}$. $4\ln(1.016)=4\times0.0159 = 0.0636$. $t=\frac{0.428}{0.0636}\approx6.73$.

Answer:

$6.73$