Consider a single nucleus of a radioactive isotope that has a decay rate equal to . The nucleus has not decayed at t 0. The probability that the nucleus will decay between time t and time t + dt is equal to a. (a) Show that this statement is consistent with the fact that the probability is 1 that the nucleus will decay between t = 0 and t = .. (Do this on paper. Your instructor may ask you to turn in this work.) (b) Show that the expected lifetime of the nucleus is equal to 1/A. Hint: The expected lifetime is equal to de-itet divided by at (Do this on paper. Your instructor may ask you to turn in this work.) (c) A sample of material contains a number of these radioactive nuclei at time t 0. What is the mean lifetime of the radioactive nuclei in the sample? (Use the following as necessary: A.)