You have to determine the age of an ancient fabric sample. You burn a small piece of fabric in excess O₂, and pass the produced CO₂ through slaked lime (Ca(OH)₂) to produce a precipitate of CaCO₃. The radioactivity of a 0.545 g sample of CaCO₃ is 2.15×10⁻⁴ Bq. Given that the radioactivity of the sample is due to ¹⁴C which has a half-life of 5715 years and the radioactivity of recent samples is 15.3 disintegrations/min per gram carbon, how old is the fabric sample?

The Chernobyl disaster took place on April 26, 1986, when reactor No. 4 in the Chernobyl Nuclear Power Plant exploded. The reactor core was exposed and burned in the air for eight days, releasing enormous amounts of radioactivity. The reactor was run on ²³⁵U and had 1.951×10⁵ kg of fuel when it exploded and the radiation levels in the vicinity were about 20,000 R/hr (Roentgen per hour). If the radiation levels are directly proportional to the amount of ²³⁵U, what will be the radiation level in the vicinity of the reactor on April 26, 2023? The half-life of ²³⁵U is 7.038×10⁸ y.

Samarium-153 is used for palliative bone pain treatment. It has a half-life of 1.93 days and decays by beta emission. If the bone of a person undergoing treatment absorbs 46% of the administered samarium, how long will it take for the concentration of the absorbed samarium to decay to 0.05% of the original amount?