the half - life of a radioactive substance is 1,550 years. what is the annual decay rate? write your answer…

Explanation:

Step1: Set up the decay formula

The general formula for exponential decay 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. For half - life, when $t = T_{1/2}$ (half - life), $A=\frac{A_0}{2}$. So, $\frac{A_0}{2}=A_0(1 - r)^{T_{1/2}}$.

Step2: Solve for the decay rate

Divide both sides of the equation $\frac{A_0}{2}=A_0(1 - r)^{T_{1/2}}$ by $A_0$ (since $A_0\neq0$), we get $\frac{1}{2}=(1 - r)^{T_{1/2}}$. Given $T_{1/2}=1550$ years, so $\frac{1}{2}=(1 - r)^{1550}$. Take the 1550th root of both sides: $1 - r=\left(\frac{1}{2}\right)^{\frac{1}{1550}}$. Then $r = 1-\left(\frac{1}{2}\right)^{\frac{1}{1550}}$.

Step3: Calculate the value of $r$

Using a calculator, $\left(\frac{1}{2}\right)^{\frac{1}{1550}}\approx0.999447$. Then $r = 1 - 0.999447=0.000553$.

Step4: Convert to percentage

To convert $r$ to a percentage, multiply by 100. So $r = 0.0553%$.

Answer:

$0.0553%$