# 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.

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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.

Calculus Concepts Cont. Therefore, the equation for the amount of a radioactive element left after time t and a positive k constant is: 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 Half-life Equation (used primarily in chemistry):

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:

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 1000- mg supply, how much will remain after 10 years? Answer:

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:

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:

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 tohalf it’s initial value Definition: Time required for something to fall to half it’s initial value Calculus Concept: A particular form of exponentialdecay Calculus Concept: A particular form of exponential decay Solve Problems: First solve forconstant of proportionality (k),then determine unknown variable Solve Problems: First solve for constant of proportionality (k), then determine unknown variable Processes of half-lives:radioactive decay,pharmaceutical science Processes of half-lives: radioactive decay, pharmaceutical science

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