Download presentation

Presentation is loading. Please wait.

Published byMarlee Mallory Modified over 2 years ago

1
AP AB Calculus: Half-Lives

2
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

3
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

4
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

5
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.

6
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.

7
Calculus Concepts Cont. Half-life Derivation Half-life Equation (used primarily in chemistry):

8
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)

9
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)

10
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:

11
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:

12
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:

13
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:

14
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

15
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

Similar presentations

OK

Exponential Growth and Decay 6.4. Exponential Decay Exponential Decay is very similar to Exponential Growth. The only difference in the model is that.

Exponential Growth and Decay 6.4. Exponential Decay Exponential Decay is very similar to Exponential Growth. The only difference in the model is that.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on oxidation and reduction worksheet Ppt on afforestation and deforestation Free download ppt on solar system Laser video display ppt on tv Ppt on solar system for grade 5 Ppt on 9-11 conspiracy theories attacks videos Ppt on review of related literature Ppt on brand marketing company Download ppt on fibonacci numbers Ppt on new york times