4 The mass of radioactive atoms in a substance can be modelled by the equation $$m = m _ { 0 } k ^ { t }$$ where \(m _ { 0 }\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.

1. 1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004 , correct to six decimal places.
   2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram.
2. The half-life of a radioactive substance is the time it takes for a mass of \(m _ { 0 }\) to reduce to a mass of \(\frac { 1 } { 2 } m _ { 0 }\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]
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