Exam (elaborations) TEST BANK FOR An introduction to Ordinary Differential Equations 1st Edition By Robinson J.C. (Solution Manual)

Exercise 1.1 Radioactive isotopes decay at random, with a ¯xed probability of decay per unit time. Over a time interval ¢t, suppose that the probability of any one isotope decaying is k¢t. If there are N isotopes, how many will decay on average over a time interval ¢t? Deduce that N(t + ¢t) ¡ N(t) ¼ ¡Nk¢t; and hence that dN=dt = ¡kN is an appropriate model for radioactive decay. Over a time interval ¢t, Nk¢t isotopes will decay. We then have N(t + ¢t) ¡ N(t) = ¡Nk¢t: Dividing by ¢t gives N(t + ¢t) ¡ N(t) ¢t = ¡Nk; and letting ¢t ! 0 we obtain, using the de¯nition of the derivative, dN dt = ¡kN: Exercise 1.2 Plutonium 239, virtually non-existent in nature, is one of the radioactive materials used in the production of nuclear weapons, and is a by-product of the generation of power in a nuclear reactor. Its half-life is approximately 24 000 years. What is the value of k that should be used in (1.1) for this isotope? Since N(t) = N(s)e¡k(t¡s), half of the isotopes decay after a time T, where N(s + T) = 1 2N(s) = N(s)e¡kT ; 1 2 1 Radioactive decay and carbon dating i.e. when 1 2 = e¡kT . Thus the half-life T = ln 2=k (as derived in Section 1.1). If T = 24000 then k = ln 2=T ¼ 2:888 £ 10¡5. Exercise 1.3 In 1947 a large collection of papyrus scrolls, including the old- est known manuscript version of portions of the Old Testament, was found in a cave near the Dead Sea; they have come to be known as the `Dead Sea Scrolls'. The scroll containing the book of Isaiah was dated in 1994 using the radiocarbon technique1; it was found to contain between 75% and 77% of the initial level of carbon 14. Between which dates was the scroll written? We have N(1994) = pN(s) = N(s)e¡k(1994¡s); where 0:75 · p · 0:77. Taking logarithms gives log p = ¡k(1994 ¡ s); and so s = 1994 + log p k : With k = 1:216 £ 10¡4 this gives (approximately) ¡372 · s · ¡155; dating the scrolls between 372 BC and 155 BC. Exercise 1.4 A large round table hangs on the wall of the castle in Winch- ester. Many would like to believe that this is the Round Table of King Arthur, who (so legend would have it) was at the height of his powers in about AD 500. If the table dates from this time, what proportion of the original carbon 14 would remain? In 1976 the table was dated using the radiocarbon tech- nique, and 91.6% of the original quantity of carbon 14 was found2. From when does the table date? If the table dates from 500 AD then we would expect N(t) = e¡k(t¡500)N(500); and so in 2003 we have N(2003) = e¡1503kN(500): The proportion of 14C isotopes remaining should there be e¡1503k ¼ 83%. 1 A.J. Jull et al., `Radiocarbon Dating of the Scrolls and Linen Fragments from the Judean Desert', Radiocarbon (1995) 37, 11{19. 2 M. Biddle, King Arthur's Round Table (Boydell Press, 2001).