justin is going to invest $90,000 and leave it in an account for 19 years. assuming the interest is…

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), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Given $P = 90000$, $A=216000$, $n = 4$ (compounded quarterly), and $t = 19$. Substitute these values into the formula: $216000=90000(1 +\frac{r}{4})^{4\times19}$.

Step2: Simplify the equation

First, divide both sides of the equation by $90000$: $\frac{216000}{90000}=(1 +\frac{r}{4})^{76}$. $2.4=(1 +\frac{r}{4})^{76}$.

Step3: Take the 76th root of both sides

Raise both sides to the power of $\frac{1}{76}$: $2.4^{\frac{1}{76}}=1+\frac{r}{4}$. Using a calculator, $2.4^{\frac{1}{76}}\approx1.0113$. So, $1.0113 = 1+\frac{r}{4}$.

Step4: Solve for $r$

Subtract $1$ from both sides: $1.0113−1=\frac{r}{4}$. $0.0113=\frac{r}{4}$. Multiply both sides by $4$ to get $r$: $r = 4\times0.0113=0.0452$. To convert to a percentage, multiply by $100$: $r = 4.52%$.

Answer:

$4.52%$