6. 0 / 7.14 points a couple plans to save for their childs college education. what principal must be…

Explanation:

Step1: Identify the 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. We are given that $A=$50000$, $r = 0.07$ (since $7%=0.07$), $n = 4$ (compounded quarterly), and $t = 18$. We need to solve for $P$.

Step2: Rearrange the formula for $P$

Starting from $A = P(1+\frac{r}{n})^{nt}$, we can solve for $P$ by dividing both sides of the equation by $(1+\frac{r}{n})^{nt}$. So, $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$.

Step3: Substitute the given values into the formula for $P$

Substitute $A = 50000$, $r=0.07$, $n = 4$, and $t = 18$ into the formula: [ \begin{align*} P&=\frac{50000}{(1+\frac{0.07}{4})^{4\times18}}\ &=\frac{50000}{(1 + 0.0175)^{72}}\ &=\frac{50000}{(1.0175)^{72}} \end{align*} ] Using a calculator, $(1.0175)^{72}\approx3.51997$. So, $P=\frac{50000}{3.51997}\approx14204.52$.

Answer:

$14204.52$