Question

A metal has half-life period of 10 days. A sample originally has a mass of 1000 mg, then the mass remaining after 50 days is

• A. \( \frac{225}{8} \) mg
• B. \( 125 \) mg
• C. \( \frac{125}{4} \) mg
• D. \( 225 \) mg

Hint:

Use the formula for half-life to find the remaining mass after a given period: \[ M(t) = M_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{\text{half}}}}. \]

Answer 1

The Correct Option is C

Step 1: Formula for remaining mass.
The formula for the remaining mass after time \( t \) given the half-life period \( T_{\text{half}} \) is: \[ M(t) = M_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{\text{half}}}} \] Where \( M_0 \) is the initial mass and \( t \) is the time elapsed.
Step 2: Apply the formula.
Given that \( M_0 = 1000 \) mg, \( T_{\text{half}} = 10 \) days, and \( t = 50 \) days, we substitute into the formula: \[ M(50) = 1000 \left( \frac{1}{2} \right)^{\frac{50}{10}} = 1000 \left( \frac{1}{2} \right)^5 = 1000 \times \frac{1}{32} = \frac{1000}{32} = \frac{125}{4} \, \text{mg} \]
Step 3: Conclusion.
Thus, the mass remaining after 50 days is \( \frac{125}{4} \) mg, corresponding to option (C).