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AP AB Calculus: Half-Lives Objective To derive the half-life equation using calculus To learn how to solve half-life problems To solve basic and challenging half-life problems To understand the applications of half-life problems in real-life Do Now: Exponential Growth Problem: In 1985, there were 285 cell phone subscribers in the town of Centerville. The number of subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994? Answer: y= a (1 + r ) ^x y= 285 (1 + .75) ^9 y= 43871 subscribers in 1994 What is a half-life? The time required for half of a given substance to decay Time varies from a few microseconds to billions of years, depending on the stability of the substance Half-lives can increase or remain constant over time Calculus Concepts Growth & Decay Derivation The rate of change of a variable y at time t is proportional to the value of the variable y at time t, where k is the constant of proportionality. dy dt ò = ky dy = y ò k dt ln y = kt + C y =e kt+C y = ekt ec y Ce kt Calculus Concepts Cont. Therefore, the equation for the amount of a radioactive element left after time t and a positive k constant is: y Ce kt The half-life of a substance is found by setting this equation equal to double the amount of substance. Calculus Concepts Cont. Half-life Derivation 2 C Ce kt ln 2 ln( e ) kt ln 2 kt ln 2 k t Half-life Equation (used primarily in chemistry): half-life ln 2 k How to solve a half-life problem Steps to solve for amount of time t Use given information to solve for k Given information: initial amount of substance (C), half of the final amount of substance (y), half-life of substance (t) Use k in the original equation to determine t Original equation: initial amount of substance (C), final amount of substance (y), constant of proportionality (k) How to solve a half-life problem Steps to solve for final amount of substance y Use given information to solve for k Given information: initial amount of substance (C), half of the final amount of substance (y), half-life of substance (t) Use k in the original equation to determine y Original equation: initial amount of substance (C), time elapsed (t), constant of proportionality (k) Basic Example #1 Problem: Suppose 10g of plutonium Pu-239 was released in the Chernobyl nuclear accident. How long will it take the 10g to decay to 1g? (Half life Pu-239 is 24,360 years.) Answer: y Ce ln . 5 t kt 1 (10) =10e24,360k 2 5 =10e24,360k ln . 5 ln e k 1 10 e 24 , 360 ln . 5 t ln . 1 ln e 24 , 360 ln . 1 24 ,360 24 , 360 k ln . 5 24 ,360 ln . 5 t t 24 ,360 ln . 1 ln . 5 t 80 ,922 . 17 years Basic Example #2 Problem: Cobalt-60 is a radioactive element used as a source of radiation in the treatment of cancer. Cobalt-60 has a half-life of five years. If a hospital starts with a 1000mg supply, how much will remain after 10 years? Answer: 10 ln . 5 y Ce 1 2 kt (1000 ) 1000 e .5 e 5 y 1000 e 2 ln . 5 5k ln . 5 5 k k 5k y 1000 e ln . 5 5 y 250 mg Challenging Example #1 Problem: The half-life of Rossidium-312 is 4,801 years. How long will it take for a mass of Rossidium-312 to decay to 98% of its original size? Answer: ln . 5 y Ce 1 kt (1) (1) e 4801 k . 98 (1) e 4801 ln . 98 k 4801 k ln . 5 4801 t 4801 2 .5 e ln . 5 t 4801 ln . 98 ln . 5 t 139 . 93 years Challenging Example #2 Problem: The half-life of carbon-14 is 5730 years. A bone is discovered which has 30 percent of the carbon-14 found in the bones of other living animals. How old is the bone? Answer: ln . 5 t y Ce 1 2 (. 3 ) . 3 e .5 e k kt 5730 k 5730 k ln . 5 5730 . 3 (1) e 5730 ln . 3 ln . 5 t 5730 t 9952 . 81 years Applications in Real Life Radioactive decay: half the amount of time for atoms to decay and form a more stable element Knowing the half-life enables one to date a partially decayed sample Examples: fossils, meteorites, carbon-14 in once-living bone and wood Biology: half the amount of time elements are metabolized or eliminated by the body Knowing the half-life enables one to determine appropriate drug dosage amounts and intervals Examples: Pharmaceutics, toxins Summary of Half-Lives Definition: Time required for something to fall to half it’s initial value Calculus Concept: A particular form of exponential decay Solve Problems: First solve for constant of proportionality (k), then determine unknown variable Processes of half-lives: radioactive decay, pharmaceutical science