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Copyright Andrew Chambers 2020. Licensed for non-commercial use only. Visit ibmathsresources.com to download the full worked mark-scheme and for 300 exploration ideas. Radioactive decay [39 marks] 1. [Maximum marks: 31] We can obtain a discrete model of radioactive decay by collectively rolling a number of dice and then after each roll removing all dice showing a six, before repeating the process with the dice left. (a) We define the number of dice left after ! rolls by ! ! . If we start with ! ! dice, find an equation for ! ! . [2] (b) By considering the relationship ! = ! !" (!) , find an equation for ! ! in the form ! ! ! !!" where ! is a constant you should find. [3] The continuous radioactive decay of atoms can be modeled with the following equation: ! ! = ! ! ! !!" ! ! : The quantity of the element remaining after time t (years). ! ! : The initial quantity of the element. !: The radioactive decay constant. (c) Carbon-14 has a half-life of 5730 years. This means that after 5730 years exactly half of the atoms of the original quantity will have decayed. Use this information to find the !, the radioactive decay constant of Carbon-14. [3] (d) You find an old manuscript and after testing the levels of Carbon-14 you find that it contains only 30% of Carbon-14 of a new piece of paper. How old is this paper? [2] (e) If we define ! ! !" = ! ! , we can evaluate improper integrals as follows: ! ! !" = lim !→! ! ! !" ! ! ! ! = lim !→! [! ! ] ! ! Show that ! ! ! !" = 1 ! ! [4]
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