Exponential Growth/Decay: $\dfrac{dN}{dt} = kN \;\Rightarrow\; N(t) = N_0 e^{kt}$
$k > 0$ for growth; $k < 0$ for decay. Half-life: $t_{1/2} = \dfrac{\ln 2}{|k|}$
Newton's Law of Cooling: $\dfrac{dT}{dt} = -k(T - T_\infty)$
Solution: $T(t) = T_\infty + (T_0 - T_\infty)\,e^{-kt}$, where $T_\infty$ = ambient temperature.
Mixture/Dilution Problems:
where $Q$ = amount of substance in tank, rate in = (concentration in)(flow rate in), rate out = $(Q/V)$ × (flow rate out).
A bacterial culture starts with 500 organisms and increases to 1{,}000 in 2 hours.
a. Find the growth rate constant $k$.
b. How many organisms are present after 5 hours?
c. How long does it take for the population to triple?
Carbon-14 has a half-life of 5{,}730 years.
a. Write the decay function $N(t)$ in terms of $N_0$.
b. A wood sample contains 30% of its original C-14. Estimate its age.
c. What percentage remains after 10{,}000 years?
A cup of coffee at 90°C cools to 70°C in 5 minutes in a room maintained at 20°C.
a. Find the cooling constant $k$.
b. Find the temperature of the coffee at $t = 10$ minutes.
c. How long does it take for the coffee to cool to 50°C?
A tank initially contains 100 L of brine with 20 kg of dissolved salt. Pure water is pumped in at 4 L/min, and the well-stirred mixture is pumped out at 4 L/min.
a. Set up and solve the DE for $Q(t)$, the amount of salt (kg) at time $t$ (min).
b. Find the salt content at $t = 25$ min.
c. How long until only 10 kg of salt remains?
