what annual interest rates r yield a balance of more than $3,000 on a 20 - year investment of $1,000? (enter…

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. Here, $P = 1000$, $A>3000$, and $t = 20$. So we have the inequality $1000(1 + r)^{20}>3000$.

Step2: Simplify the inequality

Divide both sides of the inequality $1000(1 + r)^{20}>3000$ by $1000$ to get $(1 + r)^{20}>3$.

Step3: Solve for $r$

Take the 20th root of both sides. Since the 20th - root function $y = x^{\frac{1}{20}}$ is an increasing function, we have $1 + r>3^{\frac{1}{20}}$. Then subtract 1 from both sides to find $r>3^{\frac{1}{20}}-1$.

Answer:

$r>3^{\frac{1}{20}} - 1$