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# Section 8B Doubling Time and Half-Life

Section 8B Doubling Time and Half-Life. Pages 524-535. Doubling Time and Half-Life. Exponential growth leads to repeated doubling . The time required for the quantity to double is called the doubling time .

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## Section 8B Doubling Time and Half-Life

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1. Section 8BDoubling Time and Half-Life Pages 524-535

2. Doubling Time and Half-Life Exponentialgrowth leads to repeated doubling. The time required for the quantity to double is called the doubling time. Exponential decay leads to repeated halving. The time required for the quantity to diminish by ½ is called the half-life.

3. Doubling Time Exponentialgrowth leads to repeated doubling. The time required for the quantity to double is called the doubling time. • Examples • Doubling time of magic penny was 1 day. • Doubling time for bacteria was 1 minute. • Doubling time for Powertown was ….

4. Doubling time of population is approximately 14 years.

5. Questions about Exponential Growth and Doubling Time • If the growth rate is P%, how do we determine new values? • If we know TD, how do we determine new values? • Is there a simple way to determine TD from P and vice-versa?

6. If we know doubling time, can we determine new values? Consider an initial population of 10,000 with a doubling time of 10 years.

7. If we know doubling time, can we determine new values? After a time t, the new value of the growing quantity is related to its initial value (at t = 0) by

8. ex1/525Suppose your bank account has a doubling time of 13 years. By what factor does your balance increase in 50 years? After 50 years, an exponentially growing quantity with a doublingtime of 13 years increases in size by a factor of 14.382

9. ex2/525World population doubled from 3 billion in 1960 to 6 billion in 2000. Suppose that the world population continues to grow with a doubling time of 40 years. What will the population be in 2030? After a time t, the new value of the growing quantity is related to its initial value (at t = 0) by

10. More Practice 35/533 The doubling time of a population of flies is 4 hours. By what factor does the population increase in 12 hours? In 1 week? 41/534The number of cells in a tumor doubles every 2.5 months. If the tumor begins with a single cell, how many cells will there be after 3 years? 4 years?

11. Questions about Exponential Growth and Doubling Time • If the growth rate is P %, can we determine new values? • If we know doubling time TD, can we determine new values? • Is there a simple way to determine TD from P and vice-versa?

12. Is there a simple way to determine TD from P and vice-versa? Approximate Doubling Time Formula (Rule of 70) For a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately This approximation works best for small growth rates and breaks down for growth rates over about 15%. NOTE: P is not in decimal form

13. exampleFind the approximate doubling time for the Powertown population that grows at 5% per year.

14. 47/535The CPI is increasing at a rate of 4% per year. What is its approximate doubling time? By what factor will prices increase in 3 years. TD is approximately 17.5 years. TD Formula: In 3 years, the prices will increase by a factor of approximately 1.1262. P Formula: new value = original value x (1.04)3 = original value x 1.1249. In 3 years, the prices will increase by a factor of 1.1249. NOTE: Use the formula that corresponds to the given EXACT information.

15. Questions about Exponential Growth and Doubling Time - Summary • If the growth rate is P %, can we determine a formula? • If we know doubling time TD, can we determine a formula? • Is there a simple way to determine TD from P and vice-versa? NOTE: Use the formula that corresponds to the given EXACT information.

16. Questions about Exponential Decay and Half-Life - Summary • If the decay rate is P%, how do we determine new values? • If we know the half-life TH, how do we determine new values? • Is there a way to determine TH from P and vice-versa?

17. More Practice 51*/534 The half-life of a radioactive substance is 70 years. If you start with some amount, how much will remain in 140 years? In 200 years? 57*/534Cobalt-56 has a half-life of 77 days. If you start with 1 kilogram, how much will remain after 150 days? After 1 year? 59/534 Urban encroachment is causing the area of a forest to decline at a rate of 8% per year. What is the half-life of the forest? What fraction of the forest will remain in 30 years? 61/535Poaching is causing a population of elephants to decline by 9% per year. What is the half-life for the population? If there are 10,000 elephants today, how many will remain in 50 years? NOTE: Use the formula that corresponds to the given EXACT information.

18. Homework Pages 534 - 535 # 38, 42, 49, 50, 54, 56, 60, 62 NOTE: Use the formula that corresponds to the given EXACT information.

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