question 8\n0/1 pt 3 97 details\nthe half - life of a radioactive substance is 1,598 years. what is the…

Explanation:

Step1: Recall decay formula

The decay formula for a radioactive substance is $A = A_0(1 - r)^t$, where $A$ is the final amount, $A_0$ is the initial amount, $r$ is the decay rate, and $t$ is the time. At the half - life $t = T_{1/2}$, $A=\frac{A_0}{2}$. So, $\frac{A_0}{2}=A_0(1 - r)^{T_{1/2}}$.

Step2: Simplify the equation

Divide both sides of $\frac{A_0}{2}=A_0(1 - r)^{T_{1/2}}$ by $A_0$ (since $A_0\neq0$), we get $\frac{1}{2}=(1 - r)^{T_{1/2}}$. Given $T_{1/2}=1598$ years, so $\frac{1}{2}=(1 - r)^{1598}$.

Step3: Solve for $r$

Take the 1598 - th root of both sides: $1 - r=\left(\frac{1}{2}\right)^{\frac{1}{1598}}$. Then $r = 1-\left(\frac{1}{2}\right)^{\frac{1}{1598}}$.

Step4: Calculate the value of $r$ as a percentage

First, calculate $\left(\frac{1}{2}\right)^{\frac{1}{1598}}\approx0.99945$. Then $r = 1 - 0.99945=0.00055$. As a percentage, $r = 0.0550%$.

Answer:

$0.0550%$