Radioactive Decay: Let \(N(t)\) be the mass of a radioactive element at time \(t\). The rate of change (decay) of \(N\) is proportional to its current value. \[\frac{dN}{dt}=-kN,\] where \(k > 0\) depends on the element. Solving this ODE we get \(N=ce^{-kt}\). Suppose the initial amount is \(N_0=N(0)\). Then we get \(c=N_0\). Thus the solution is \[N(t)=N_0e^{-kt}.\] Carbon Dating: There are 2 types of carbon atoms: \(^{12}C\) (the stable nuclide) and radioactive \(^{14}C\) with a halflife of about 5,730 years. The ratio \(^{14}C: ^{12}C\) is approximately constant in nature. A living creature taking carbon from nature is made up with this ratio. After its death \(^{14}C\) starts to decay.

Example. Suppose \(25\%\) of the original amount of \(^{14}C\) remained in a fossil. Find the age of the fossil.

Solution. First we find \(k\) for \(^{14}C\) in \(N(t)=N_0e^{-kt}\). We know that \(\frac{N_0}{2}=N_0e^{-5730k}\). Solving we get \(k=\ln 2/5730\) and hence \[N(t)=N_0e^{-t\ln 2/5730}.\] For the fossil we want to find \(t\) for which \(N(t)=25N_0/100\). Plugging this into the preceding equation we get \[25N_0/100=N_0e^{-t\ln 2/5730} \implies t=11,460 \text{ years}\] The answer is not surprising because of the halflife of \(^{14}C\) (about 5,730 years).

Newton's Law of Cooling: Let \(T(t)\) be the temperature of an object at time \(t\). Then \[\frac{dT}{dt}=-k(T-T_a),\] where \(k > 0\) is a constant and \(T_a\) is the constant ambient temperature. Solving this ODE we get \(T=T_a+ce^{-kt}\). Suppose the initial temperature is \(T_0=T(0)\). Then we get \(c=T_0-T_a\). Thus the solution is \[T(t)=T_a+(T_0-T_a)e^{-kt}.\] Note that \(T(t)=T_a+(T_0-T_a)e^{-kt}\to T_a\) as \(t\to \infty\).

Kirchhoff's Circuit Law: Consider an electric circuit with a capacitor, resistor, and battery. Let \(Q(t)\) be the charge of the capacitor at time \(t\). Then \[R\frac{dQ}{dt}+\frac{Q}{C}=V,\] where \(R\) is the constant resistant, \(C\) is the constant capacitance, and \(V\) is the constant voltage supplied. Solving this ODE we get \(Q=CV+ke^{-t/RC}\). Suppose the initial charge is \(Q(0)=0\). Then we get \(k=-CV\). Thus the solution is \[Q(t)=CV(1-e^{-t/RC}).\] Note that \(Q(t)=CV(1-e^{-t/RC})\to CV\) as \(t\to \infty\).

Mixing Rate Problems: The key idea for mixing rate problems is that the effective rate is the rate in minus the rate out.

Example. Suppose a tank contains 200 gal of salt water containing 100 lb salt. Then salt water with 3 lb salt/gal is pumped into the tank at a rate \(2\) gal/min and the "well-stirred" salt water is pumped out of the tank at a rate \(2\) gal/min. Suppose \(Q(t)\) denotes the amount of salt in lb at time \(t\) min. Then \[\frac{dQ}{dt}=2\cdot 3-2\cdot \frac{Q}{200},\, Q(0)=100\] Solving this ODE we get \(Q=600+ce^{-t/100}\). Using the initial condition \(Q(0)=100\) we get \(c=-500\). Thus the solution is \[Q(t)=600-500e^{-t/100}.\] Note that \(Q(t)=600-500e^{-t/100}\to 600\) as \(t\to \infty\).

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