Doubling Time and Half-Life Unit 8B Doubling Time and Half-Life Ms. Young

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. Ms. Young

Doubling Time After a time t, an exponentially growing quantity with a doubling time of Tdouble 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 2t/Tdouble Ms. Young

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 Tdouble 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 250 yr/40 yr = 6 billion x 21.25 ≈ 14.3 billion Ms. Young

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%. Work through several examples comparing perhaps two different countries and their growth rates or two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.” Ms. Young

Exponential Decay and Half-Life After a time t, an exponentially decaying quantity with a half-life time of Thalf 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/Thalf Ms. Young

The Approximate Half-Life Formula 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%. Work through several examples comparing perhaps two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.” Ms. Young

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 an exponentially decaying quantity, use a negative value for r. The halving time is Ms. Young