SECTION 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 357 FUN FACT Uranium is a radionuclide that has an extremely long half-life. Naturally occurring uranium-238 present in the Earth’s crust has a half-life of almost 4.5 billion years. If you take a soil sample anywhere in the world, including your own backyard, you will find uranium atoms that date back to when the Earth was formed. j Solution (a) Using formula (2), the number N of cells at time t is ( ) = N t N ekt 0 where N0 is the initial number of bacteria present and k is a positive number. (b) To find the growth rate k, note that the number of cells doubles in 3 hours, so ( ) = N N 3 2 0 ( ) = = = = = ≈ ⋅ ⋅ N N e N e N e k k Since 3 , 2 2 3 ln2 1 3 ln2 0.23105 k k k 0 3 0 3 0 3 Divide both sides by N . 0 Write the exponential equation as a logarithm. The function that models this growth process is therefore ( ) = N t N e t 0 0.23105 (c) The time t needed for the size of the colony to triple requires that = N N3 .0 Substitute N3 0 for N to get = = = = ≈ N N e e t t 3 3 0.23105 ln3 ln3 0.23105 4.755 hours t t 0 0 0.23105 0.23105 It will take about 4.755 hours, or 4 hours and 45 minutes, for the size of the colony to triple. (d) If a population doubles in 3 hours, it will double a second time in 3 more hours, for a total time of 6 hours. 2 Model Populations That Obey the Law of Uninhibited Decay Radioactive materials follow the law of uninhibited decay. Uninhibited Radioactive Decay The amount A of a radioactive material present at time t is given by ( ) = < A t A e k 0 kt 0 (3) where A0 is the original amount of radioactive material and k is a negative number that represents the rate of decay. All radioactive substances have a specific half-life , which is the time required for half of the radioactive substance to decay. Carbon dating uses the fact that all living organisms contain two kinds of carbon, carbon-12 (a stable carbon) and carbon-14 (a radioactive carbon with a half-life of 5730 years). While an organism is living, the ratio of carbon-12 to carbon-14 is constant. But when an organism dies, the original amount of carbon-12 present remains unchanged, whereas the amount of carbon-14 begins to decrease. This change in the amount of carbon-14 present relative to the amount of carbon-12 present makes it possible to calculate when the organism died.
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