AP AB Calculus:Half-Lives

ObjectiveTo 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 decayTime 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 ConceptsGrowth & 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.

ktCey

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.

ktCey

Calculus Concepts Cont.Half-life

Derivation

ln 2 ktln 2 tk

ln 2half-lifek

Half-life Equation (used primarily in chemistry):

ktCeC 2)ln(2ln kte

How to solve a half-life problem

Steps to solve for amount of time tUse 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 kGiven 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 #1Problem: 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:

ktCey

ke 360,24ln5.ln

360,245.ln

k

360,245.ln

101t

e

360,245.ln

ln1.lnt

e

360,245.ln1.ln t

5.ln1.ln360,24

t

yearst 17.922,80

Basic Example #2Problem: 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: ktCey

mgy 250

ke51000)1000(21

ke55.

k55.ln

55.ln

k

55.ln10

1000ey 5.ln21000ey

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:

ktCey ke4801)1()1(

21

ke48015.

48015.ln

k

48015.ln

)1(98. e

t48015.ln98.ln

5.ln98.ln4801

t

yearst 93.139

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:ktCey

yearst 81.9952

ke57303.)3(.21

ke57305.

57305.ln

k

57305.ln

)1(3.t

e

57305.ln3.ln t

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 sampleExamples: 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 intervalsExamples: 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 variableProcesses of half-lives: radioactive decay, pharmaceutical science