Math 1B Quiz 11 Thursday 15 November 2007 GSI Theo Johnson Freyd http math berkeley edu theojf Name 1 3 pts Solve the following initial value problem to write f as a function of t df t f2 1 dt f 2 0 This equation is separable so we multiply and divide and then plug in our initial value df dt df 1 f2 t f2 1 t dt t2 C 2 t2 f tan C 2 22 arctan 0 C 2 0 2 C arctan f C 2 f t tan 1 t2 2 2 2 3 pts Solve the following differential equation to find a one parameter family of solutions for y as a function of x 2xy 0 y 6x We use the method of finding a multiplier in order to recognize the equation as a separable one 1 y0 y 3 Z 2x dx 1 ln x 2x 2 1 x e 2 ln x 1 0 x y0 y xy 3 x 2x x y 2x3 2 C y 2x C x 2 3 4 pts Carbon has two stable isotopes carbon 12 12 C and carbon 13 13 C and one relatively common radioactive isotope carbon 14 14 C produced in the upper atmosphere by bombardment with cosmic radiation Plants absorb atmospheric carbon and hence the concentration of 14 C in plants is equal to the atmospheric concentration When plants die they do not absorb any new carbon The amount of 14 C in archeological samples is used to date archeological sites a Like all radioactive materials 14 C decays at a constant relative rate the amount that decays in any given period of time is proportional to the amount present Write a differential equation modeling the amount of 14 C in a given amount of time Let f t be the amount of 14 C after time t Then df kf dt We write k for the coefficient since we know that the amount of 14 C is decreasing and this lets us use a positive k Remark the solution to this differential equation is f t f 0 e kt b The half life of 14 C is 5730 years and the atmospheric concentration of 14 C is 600 billion atoms per mole roughly one part per trillion What is the solution to your differential equation relating how much time has elapsed with the amount of 14 C left You do not need to simplify but you do need to use units We have f t f 0 e kt Before any 14 C has decayed i e at time t 0 it is at f 0 600 billion atoms per mole In 5730 years the total 14 C has halved 1 2 e k 5730years so k ln 2 5730 Thus f t 600 109 rmatoms mole e ln 2 5730 yrs t c A sample from Fell s Cave in southern Chile has a 14 C concentration of 150 billion atoms per mole Roughly what is the date of the archeological site By solving the equation above for f t 150 billion atoms per mole we get the right t Faster for the 14 C to decay from 600 billion atoms per mole to 150 billion atoms per mole requires two halvings thus two half lifes 11460 years have passed so the sample is from roughly 9000 B C E 3 d What is the rate of radioactive decay of 14 C 12 C in the sample if the current concentration of 14 C is 150 billion atoms per mole You do not need to simplify but you do need to report units We recall that f 0 t kf t Then if f t 150 billion atoms per mole we must have f 0 t ln 2 5730 yrs 150 109 atoms mole 4
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