Question

the half - life, t, for a particular radioactive element is 22 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 - rate formula

The formula for the half - life $T$ of a radioactive substance is $T=\frac{\ln 2}{k}$, where $k$ is the decay rate. We need to solve for $k$.

Step2: Rearrange the formula for $k$

From $T = \frac{\ln 2}{k}$, we can cross - multiply to get $kT=\ln 2$. Then $k=\frac{\ln 2}{T}$.

Step3: Substitute the given value of $T$

Given $T = 22$ min, so $k=\frac{\ln 2}{22}$.

Step4: Convert $k$ to a percentage

To convert $k$ to a percentage, we multiply by 100. So the percentage decay rate $r = 100k=100\times\frac{\ln 2}{22}$.
Calculate $100\times\frac{\ln 2}{22}=\frac{100\times0.693147}{22}\approx 3.15067$.

Answer:

$3.2$