justin invested $7,300 in an account paying an interest rate of 1.5% compounded quarterly. assuming no…

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 $P = 7300$, $A = 8560$, $r=0.015$ (since $1.5%=0.015$), and $n = 4$ (compounded quarterly). Substitute these values into the formula: $8560=7300(1 +\frac{0.015}{4})^{4t}$.

Step2: Simplify the equation

First, divide both sides of the equation by $7300$: $\frac{8560}{7300}=(1 + 0.00375)^{4t}$. $\frac{8560}{7300}=1.00375^{4t}$. $1.1726=(1.00375)^{4t}$.

Step3: Take the natural logarithm of both sides

$\ln(1.1726)=\ln(1.00375^{4t})$. Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we get: $\ln(1.1726)=4t\ln(1.00375)$.

Step4: Solve for $t$

We know that $\ln(1.1726)\approx0.160$ and $\ln(1.00375)\approx0.00374$. So, $0.160 = 4t\times0.00374$. $4t=\frac{0.160}{0.00374}\approx42.8$. $t=\frac{42.8}{4}=10.7$.

Answer:

$10.7$ years