Exponential Growth
The Exponential Differential Equation An equation is called a differential equation if it is an equation that contains derivatives. In this section, we will consider differential equations of the type dy/dt = ky This differential equation can be interpreted as the equation that models the following statement, "The rate of growth is proportional to the amount present." Typically, solutions to differential equations are highly elusive. However this differential equation has solution that are quite nice.
Proof Let y = Ce^{kt} then y' = kCe^{kt} = ky
Radioactive Decay A fundamental principle of nuclear chemistry states that the rate of decay of a radioactive element is proportional to the amount present. In other words, the more you have, the more there is to lose. Mathematically, if y is the amount present at time t, then y' = ky From the above statement, this differential equation has a solution y = Ce^{kt} for some constants C and k. Notice that when we plug in 0, y(0) = C Hence, the constant C corresponds to the initial amount present.
Example This year 500 pounds of Plutonium 239 (nuclear waste) was dumped in the Nevada desert. The half life of Pu239 is 24,360 years. How long will it take until there is only 1 pound left?
Solution Since y' = ky we can conclude that y = Ce^{kt} The initial amount present is 500 pounds, hence C = 500 The half life is 24,360 years so that y = 250 when t = 24360. Plugging in gives 250 = 500e^{k24,360} 1/2 = e^{k24,360} ln(0.5) = 24,360k
ln 0.5 Hence
We set
and solve for t to get t = 218,407 years.
Exercise Suppose that there is a fruit fly infestation in the central valley. Being an environmentalist, you propose a plan to spread 50,000 infertile fruit flies in the area to control the situation. Presently, you have in your laboratory 1,000 fruit flies. In 1 week they will reproduce to a population of 3,000 fruit flies. The farmers want to know when you will be ready to drop your infertile fruit flies. What should you tell them? Hold your mouse on the yellow rectangle for the answer.
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