Radium decays according to the function \[ Q(t)=Q_{0} e^{-k t} \] where \( Q \) represents the quantity remaining after \( t \) years and \( k \) is the decay constant 0.00043 . What is the half-life of radium? Calculate how long it will take for 120 g of radium to decay to 60 g . Round your answer to the nearest year. A. 12 years B. 806 years C. 1612 years D. 55,813 years

To find the half-life of radium, we set \( Q(t) = \frac{1}{2} Q_0 \) in the decay equation: \[ \frac{1}{2} Q_0 = Q_0 e^{-kt} \] Dividing both sides by \( Q_0 \): \[ \frac{1}{2} = e^{-kt} \] Taking the natural logarithm of both sides: \[ \ln\left(\frac{1}{2}\right) = -kt \] Solving for \( t \): \[ t = \frac{\ln(2)}{k} \approx \frac{0.693}{0.00043} \approx 1612 \text{ years} \] Now, to calculate how long it will take for 120 g of radium to decay to 60 g, we use the same function: \[ 60 = 120 e^{-0.00043 t} \] Dividing by 120: \[ 0.5 = e^{-0.00043 t} \] Following similar steps as before: \[ \ln(0.5) = -0.00043 t \] Therefore: \[ t = \frac{\ln(2)}{0.00043} \approx 1612 \text{ years} \] So, the correct answer is **C. 1612 years**. --- Radium was discovered in 1898 by Marie Curie and her husband Pierre while they were conducting research on radioactivity. This historic finding not only opened doors to groundbreaking studies in nuclear physics and medicine but also raised concerns about health effects, leading to better understanding of radiation safety. In real-world applications, radium has been used in medical treatments, particularly in brachytherapy for cancer. However, because of its harmful effects, many safer alternatives have emerged, making knowledge of radium's decay and properties crucial for safe utilization in contemporary practices.