Radioactive Decay Half-Life Calculation

Steps:

• Count rate follows exponential decay: final rate = initial rate × (1/2)^(t / T), where T is half-life.
• 500 = 4000 × (1/2)^(72 / T).
• Simplify: 500/4000 = 1/8 = (1/2)^3, so 72 / T = 3.
• T = 72 / 3 = 24 minutes.

Why D is correct:

• A factor of 8 reduction equals three half-lives (2^3 = 8), so half-life is 72/3 = 24 minutes per the decay law.

Why the others are wrong:

• A: 8 minutes implies 9 half-lives (72/8), reducing by 2^9 = 512 (to ~8 cpm), excessive.
• B: 9 minutes implies 8 half-lives (72/9), reducing by 2^8 = 256 (to ~16 cpm), excessive.
• C: 18 minutes implies 4 half-lives (72/18), reducing by 2^4 = 16 (to 250 cpm), insufficient.

Final answer: D