A teacher supplies averaged detector rates, so fractional counts/min are possible. A separate background run gives a constant mean 60 counts/min. The source-plus-background (time in min, mean rate in counts/min) pairs are (0, 140), (7, 100), (14, 80), (21, 70). A student claims half-life = 7 min. Subtract background, test the claimed model against every reading, and calculate the model's expected total rate at 28 min. Use agreement within 5% of the predicted net rate as a classroom comparison rule. Explain one limitation of this test. No radioactive source is handled by students.

Question

Accepted Answer

Net rates 80, 40, 20, 10 counts/min. The claimed 7 min half-life is consistent with the stated rule. Its expected total rate at 28 min is 65 counts/min. The detector receives both source and background events. Subtract 60 counts/min from every total, giving source-only means 80, 40, 20, 10 counts/min; background must not be halved with the source. If the proposed half-life is 7 min, successive predicted source rates are 80, 40, 20, 10 counts/min. Compare each corrected measurement with its own predicted value, not with half the original gross rate. The relative departures from this model are 0, 0, 0, 0%. The dataset therefore passes the supplied 5% consistency rule. After four proposed half-lives, the predicted source rate is 80/16 = 5 counts/min. Add the unchanged background to predict a total 65 counts/min. This is a model forecast, not a guaranteed individual count. Radioactive events fluctuate randomly, and the background estimate also has uncertainty. Longer counting intervals, repeat averages and a check for changing geometry would strengthen the test; the arbitrary 5% classroom rule is not a statistical confidence calculation.

A source close to background

What a complete response should do

Recorded total detector rate

How did this one feel?

Recorded total detector rate — supplied values
Time (min)	Mean detector rate (counts/min)
0	140
7	100
14	80
21	70