The bombardment of the upper atmosphere by cosmic

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The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioac- tive isotope 14 C of carbon with a half-life of about 5730 years. Vegetation absorbs car- bon dioxide through the atmosphere and ani- mal life assimilates 14 C through food chains. When a plant or animal dies it stops replac- ing its carbon and the amount of 14 C begins to decrease through radioactive decay. There- fore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 30% as much 14 C radioactivity as does plant material on Earth today. Estimate the age of the parchment. 1. t = 9953 years correct 2. t = 9923 years 3. t = 9956 years 4. t = 9967 years 5. t = 9945 years Explanation: Let y ( t ) be the level of radioactivity. Thus, y ( t ) = y (0) e kt and k is determined by using the half-life: y (5730) = 1 2 y (0) y (0) e k (5730) = 1 2 y (0) e 5730 k = 1 2 5730 k = ln 1 2 k = ln 1 2 5730 = ln 2 5730 . If 30% of the 14 C remains, then we know that y ( t ) = 0 . 3 y (0) 0 . 3 = e t (ln2) / 5730 ln 0 . 3 = t ln 2 5730 t = 5730(ln 0 . 3) ln 2 9953 .

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