2. Copyright © 2011 Pearson Education, Inc. Exponential Astonishment

3. Copyright © 2011 Pearson Education, Inc. Slide 8-3 Unit 8B Doubling Time and Half-Life

4. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-4 Doubling and Halving Times The time required for each doubling in exponential growth is called doubling time. The time required for each halving in exponential decay is called halving time.

5. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-5 After a time t, an exponentially growing quantity with a doubling time of T double increases in size by a factor of. The new value of the growing quantity is related to its initial value (at t = 0) by New value = initial value x 2 t/T double Doubling Time

6. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-6 Example World Population Growth: World population doubled from 3 billion in 1960 to 6 billion in 2000. Suppose that the world population continued to grow (from 2000 on) with a doubling time of 40 years. What would be the population in 2050? The doubling time is T double 40 years. Let t = 0 represent 2000 and the year 2050 represent t = 50 years later. Use the 2000 population of 6 billion as the initial value. new value = 6 billion x 2 50 yr/40 yr = 6 billion x 2 1.25 ≈ 14.3 billion

7. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-7 Approximate Double Time Formula (The 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%.

8. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-8 After a time t, an exponentially decaying quantity with a half-life time of T half decreases in size by a factor of. The new value of the decaying quantity is related to its initial value (at t = 0) by New value = initial value x (1/2) t/T half Exponential Decay and Half-Life

9. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-9 For a quantity decaying exponentially at a rate of P% per time period, the half-life is approximately This approximation works best for small decay rates and breaks down for decay rates over about 15%. The Approximate Half-Life Formula

10. 8-B Copyright © 2011 Pearson Education, Inc. Slide 8-10 Exact Doubling Time and Half-Life Formulas For more precise work, use the exact formulas. These use the fractional growth rate, r = P/100. For an exponentially growing quantity, the doubling time is For a exponentially decaying quantity, use a negative value for r. The halving time is