select the correct answer.\ncurium - 243 has a half - life of 28.5 days. in a sample of 5.6 grams of curium…

Explanation:

Step1: Recall the radioactive - decay formula

The formula for radioactive decay is $N = N_0\left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}$, where $N$ is the amount remaining, $N_0$ is the initial amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life. Here, $N_0 = 5.6$ grams, $t = 12$ days, and $T_{1/2}=28.5$ days.

Step2: Substitute the values into the formula

$N = 5.6\times\left(\frac{1}{2}\right)^{\frac{12}{28.5}}$ First, calculate the exponent: $\frac{12}{28.5}\approx0.421$. Then, find $\left(\frac{1}{2}\right)^{0.421}$. Let $y=\left(\frac{1}{2}\right)^{0.421}$, then $\ln y = 0.421\times\ln\left(\frac{1}{2}\right)=0.421\times(-\ln 2)\approx0.421\times(- 0.693)= - 0.292$. So, $y = e^{-0.292}\approx0.747$. Now, $N = 5.6\times0.747\approx4.18$ grams.

Answer:

C. 4.18 grams