Copyright © 2011 Pearson Education, Inc. Exponential Astonishment

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

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.

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

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 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 ≈ 14.3 billion

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-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

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

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