exponential growth or decay.
The length of time required for a quantity that is decaying exponentially to be reduced by half is
called its half-life.
EXAMPLE 12
The population of a country is growing at a rate proportional to its population. If the growth rate
per year is 4% of the current population, how long will it take for the population to double?
SOLUTION: If the population at time t is P, then we are given that Substituting in
equation (2), we see that the solution is
P = P 0 e0.04t,
where P 0 is the initial population. We seek t when P = 2P 0 :EXAMPLE 13
The bacteria in a certain culture increase continuously at a rate proportional to the number
present.
(a) If the number triples in 6 hours, how many will there be in 12 hours?
(b) In how many hours will the original number quadruple?
SOLUTIONS: We let N be the number at time t and N 0 the number initially. Thenhence, C = ln N 0. The general solution is then N = N 0 ekt, with k still to be determined.
Since N = 3N 0 when t = 6, we see that 3N 0 = N 0 e^6 k and that ln 3. Thus
N = N 0 e(t ln 3)/6.
(a) When t = 12, N = N 0 e2 ln 3 = N 0 eln 3^2 = N 0 eln 9 = 9N 0.
(b) We let N = 4N 0 in the centered equation above, and getEXAMPLE 14
Radium-226 decays at a rate proportional to the quantity present. Its half-life is 1612 years. How
long will it take for one quarter of a given quantity of radium-226 to decay?
SOLUTION: If Q(t) is the amount present at time t, then it satisfies the equation