To determine the count rate of a radioactive material after a certain period, we can use the concept of half-life. The half-life is the time required for half of the radioactive atoms in a sample to decay. In this case, we are given: - Initial count rate: 800 counts/min - Half-life: 4 days - Time elapsed: 16 days ### Step-by-Step Explanation 1. **Understanding Half-Life**: The half-life of a radioactive substance is the time it takes for half of the radioactive atoms to decay. After each half-life, the amount of the substance (or the count rate in this case) is halved. 2. **Calculating the Number of Half-Lives**: To find out how many half-lives fit into the 16 days, we divide the total time by the half-life: \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{16 \text{ days}}{4 \text{ days}} = 4 \] This means that 4 half-lives have passed in 16 days. 3. **Calculating the Count Rate After Each Half-Life**: We start with an initial count rate of 800 counts/min. After each half-life, the count rate is halved: - After 1st half-life (4 days): \[ \text{Count rate} = \frac{800}{2} = 400 \text{ counts/min} \] - After 2nd half-life (8 days): \[ \text{Count rate} = \frac{400}{2} = 200 \text{ counts/min} \] - After 3rd half-life (12 days): \[ \text{Count rate} = \frac{200}{2} = 100 \text{ counts/min} \] - After 4th half-life (16 days): \[ \text{Count rate} = \frac{100}{2} = 50 \text{ counts/min} \] 4. **Final Count Rate**: After 16 days, the count rate of the radioactive material is 50 counts/min. ### Conclusion The correct answer is **C. 50 count/min**. ### Explanation of Other Options: - **A. 200 count/min**: This would be the count rate after 2 half-lives (8 days), not 16 days. - **B. 100 count/min**: This would be the count rate after 3 half-lives (12 days), not 16 days. - **D. 25 count/min**: This would be the count rate after 5 half-lives (20 days), which is beyond the 16 days we are considering. ### Common Pitfalls: - **Miscounting Half-Lives**: Ensure you correctly calculate the number of half-lives that fit into the total time. - **Forgetting to Halve the Count Rate**: Each half-life requires you to halve the previous count rate; missing this step can lead to incorrect answers. ### Revision Summary: - The half-life is the time taken for half of the radioactive material to decay. - Count rate decreases by half for each half-life that passes. - For 16 days with a half-life of 4 days, 4 half-lives occur. - The final count rate after 16 days is 50 counts/min.