approximately how much principal would need to be placed into an account earning 3.575% interest compounded…

Explanation:

Step1: Identify compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the accumulated amount, $P$ is the principal, $r$ 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. We need to solve for $P$, so $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$.

Step2: Convert given values to appropriate form

The annual interest rate $r = 3.575%=0.03575$, the number of times compounded per year $n = 4$ (quarterly compounding), the number of years $t = 30$, and the accumulated amount $A=$68000$.

Step3: Substitute values into the formula

[ \begin{align*} P&=\frac{68000}{(1+\frac{0.03575}{4})^{4\times30}}\ &=\frac{68000}{(1 + 0.0089375)^{120}}\ &=\frac{68000}{(1.0089375)^{120}} \end{align*} ] Using a calculator, $(1.0089375)^{120}\approx2.8684$. Then $P=\frac{68000}{2.8684}\approx23706$.

Answer:

a. $$23,706$