Question

the half - life, t for a particular radioactive element is 4 min. find the decay rate of the element. the decay rate is % per min. (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Recall decay - formula

The formula for radioactive decay is $A = A_0e^{-rt}$, where $A$ is the amount of the substance at time $t$, $A_0$ is the initial amount, $r$ is the decay rate, and $t$ is the time. At the half - life $T$, $A=\frac{A_0}{2}$.

Step2: Substitute values into formula

Substitute $A = \frac{A_0}{2}$ and $t = T$ into $A = A_0e^{-rt}$. We get $\frac{A_0}{2}=A_0e^{-rT}$. Since $A_0
eq0$, we can divide both sides of the equation by $A_0$ to obtain $\frac{1}{2}=e^{-rT}$.

Step3: Solve for $r$

Given $T = 4$ min, we have $\frac{1}{2}=e^{-4r}$. Take the natural logarithm of both sides: $\ln(\frac{1}{2})=\ln(e^{-4r})$. Using the property $\ln(e^x)=x$, we get $\ln(\frac{1}{2})=-4r$. Since $\ln(\frac{1}{2})=-\ln(2)$, then $-\ln(2)=-4r$. Solving for $r$, we have $r=\frac{\ln(2)}{4}$.

Step4: Convert $r$ to percentage

To convert $r$ to a percentage, multiply $r$ by 100. So the percentage decay rate $R = 100r=\frac{100\ln(2)}{4}=25\ln(2)$.

Answer:

$17.3$%