I'm slightly obsessed with mathematics, my cats, and the Dave Matthews Band.
The problem is as follows:
"Plutonium-239 has a decay rate of approximately 0.003% per year. Suppose that plutonium-239 is released into the atmosphere each year for 20 years at a constant rate of 1 lb per year. How much plutonium-239 will remain in the atmosphere after 20 years?"
The amount of plutonium remaining is given exactly by the definite integral shown, where the chemical is released into the atmosphere at a rate of R(t) pounds per year for T years at a decay rate of k.
∫_(t to 0)(R(t) e^kt (dt)
Substitute the given values in the definition for the future amount. The value of k will be negative because it is decaying.
∫(20 to 0) 1 e^(-0.003t)(dt)
Now, we integrate with respect to t.
[-333.33e^(-0.003(t)) ](20 to 0)
Lastly, we evaluate the result over the integral from t = 0 from t = 20.
-333.33(e^(-0.003(20))-e^(-0.003(0))) ≈ 19.41162802
Therefore, the approximate amount of plutonium-239 in the atmosphere after 20 years is 19.412 lbs.
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