a radioactive element has a half - life of six days. what is the approximate decay rate of the element after…

Explanation:

Step1: Recall decay - formula

The formula for radioactive decay is $N = N_0(1 - r)^t$, where $N$ is the amount of the substance at time $t$, $N_0$ is the initial amount, $r$ is the decay rate, and $t$ is the time. For a half - life $T$, when $t = T$, $N=\frac{N_0}{2}$. Given $T = 6$ days, we have $\frac{N_0}{2}=N_0(1 - r)^6$.

Step2: Solve for $(1 - r)$

Divide both sides of the equation $\frac{N_0}{2}=N_0(1 - r)^6$ by $N_0$ (since $N_0\neq0$), we get $\frac{1}{2}=(1 - r)^6$. Then, $1 - r=\left(\frac{1}{2}\right)^{\frac{1}{6}}$.

Step3: Calculate $r$

First, find $\left(\frac{1}{2}\right)^{\frac{1}{6}}=2^{-\frac{1}{6}}\approx0.891$. Then, $r = 1-(1 - r)=1 - 2^{-\frac{1}{6}}\approx1 - 0.891=0.109$ or $10.9%$.

Answer:

10.9%