find the interest rate to the nearest hundredth of a percent that will produce $3500, if $2500 is left at…

find the interest rate to the nearest hundredth of a percent that will produce $3500, if $2500 is left at interest compounded quarterly for 4.5 yr. use the formula $a = p(1+\frac{r}{n})^{nt}$ for compound interest.\nthe interest rate to the nearest hundredth of a percent is $square$%.\n(do not round until the final answer. then round to the nearest hundredth as needed.)

find the interest rate to the nearest hundredth of a percent that will produce $3500, if $2500 is left at interest compounded quarterly for 4.5 yr. use the formula $a = p(1+\frac{r}{n})^{nt}$ for compound interest.\nthe interest rate to the nearest hundredth of a percent is $square$%.\n(do not round until the final answer. then round to the nearest hundredth as needed.)

Answer

Explanation:

Step1: Identify values in the formula

We have $A = 3500$, $P=2500$, $n = 4$ (compounded quarterly), and $t = 4.5$. Substitute into $A=P(1 +\frac{r}{n})^{nt}$. So, $3500=2500(1+\frac{r}{4})^{4\times4.5}$.

Step2: Simplify the equation

First, divide both sides by 2500: $\frac{3500}{2500}=(1 +\frac{r}{4})^{18}$. Which simplifies to $1.4=(1+\frac{r}{4})^{18}$.

Step3: Take the 18 - th root of both sides

$1.4^{\frac{1}{18}}=1+\frac{r}{4}$. Calculate $1.4^{\frac{1}{18}}\approx1.01937$. So, $1.01937 = 1+\frac{r}{4}$.

Step4: Solve for r

Subtract 1 from both sides: $1.01937-1=\frac{r}{4}$, which gives $0.01937=\frac{r}{4}$. Then multiply both sides by 4: $r = 4\times0.01937=0.07748$.

Step5: Convert to percentage and round

Multiply by 100 to get the percentage: $r = 7.748%$. Rounding to the nearest hundred - th, $r\approx7.75%$.

Answer:

$7.75$