the half - life of strontium - 90 is approximately 29 years. how much of a 500 g sample of strontium - 90…

Explanation:

Step1: Determine the decay - constant formula from half - life

The half - life formula is $T_{1/2}=\frac{\ln 2}{r}$, where $T_{1/2}$ is the half - life and $r$ is the decay constant. Given $T_{1/2} = 29$ years, we can solve for $r$: $r=\frac{\ln 2}{T_{1/2}}=\frac{\ln 2}{29}$.

Step2: Identify the initial amount and time

The initial amount $I = 500$ g and the time $t = 58$ years.

Step3: Use the decay formula

The decay formula is $A=I(1 - r)^t$. Substituting $r=\frac{\ln 2}{29}$, $I = 500$, and $t = 58$ into the formula. First, note that since $t = 58$ years and $T_{1/2}=29$ years, $t = 2T_{1/2}$. We know that after each half - life, the amount of the substance is halved. After the first 29 years, the amount is $\frac{500}{2}=250$ g, and after another 29 years (total of 58 years), the amount is $\frac{250}{2}=125$ g. Mathematically, using the formula $A = I(1 - r)^t$, with $r=\frac{\ln 2}{29}$ and $t = 58$: [ \begin{align*} A&=500\left(1-\frac{\ln 2}{29}\right)^{58}\ &=500\times\left(\frac{1}{2}\right)^2\ &=125 \end{align*} ]

Answer:

125 g