E The radioactive substance uranium-240 has a half-life of 14 hours. The amount A (t) of a sample of uranium-240 remaining (in grams) after t hours is given by the following exponential function. 1 14 A (t) = 4700 ( Find the initial amount in the sample and the amount remaining after 50 hours. Round your answers to the nearest gram as necessary. Initial amount: grams Amount after 50 hours: grams Х 6

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.