5. what rate would you need to have so that your $3,000 grows to $5,000 in 10 years?

Explanation:

Step1: Use compound - interest formula

The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the number of years. In this case, we assume annual compounding. We know that $A = 5000$, $P = 3000$, and $t = 10$. So, $5000=3000(1 + r)^{10}$.

Step2: Rearrange the equation

First, divide both sides of the equation by 3000: $\frac{5000}{3000}=(1 + r)^{10}$, which simplifies to $\frac{5}{3}=(1 + r)^{10}$.

Step3: Take the 10 - th root of both sides

$1 + r=\left(\frac{5}{3}\right)^{\frac{1}{10}}$.

Step4: Solve for $r$

$r=\left(\frac{5}{3}\right)^{\frac{1}{10}}-1$. Using a calculator, $\left(\frac{5}{3}\right)^{\frac{1}{10}}\approx1.0524$, so $r\approx1.0524 - 1=0.0524$ or $5.24%$.

Answer:

Approximately $5.24%$