A teacher supplies averaged detector rates, so fractional counts/min are possible. A separate background run gives a constant mean 100 counts/min. The source-plus-background (time in min, mean rate in counts/min) pairs are (0, 500), (5, 300), (10, 200), (15, 150). A student claims half-life = 5 min. Subtract background, test the claimed model against every reading, and calculate the model's expected total rate at 20 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 400, 200, 100, 50 counts/min. The claimed 5 min half-life is consistent with the stated rule. Its expected total rate at 20 min is 125 counts/min. The detector receives both source and background events. Subtract 100 counts/min from every total, giving source-only means 400, 200, 100, 50 counts/min; background must not be halved with the source. If the proposed half-life is 5 min, successive predicted source rates are 400, 200, 100, 50 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 400/16 = 25 counts/min. Add the unchanged background to predict a total 125 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.

Background becomes important

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	500
5	300
10	200
15	150