Growth Result As A Function of Time

What is a growth and decay model?

Simple growth and decay are mathematical models used to describe how quantities change over time. Growth equations involve a constant multiplier that increases the current value, while decay equations use a constant that decreases it. For instance, population growth often exhibits exponential behavior, where the rate of change is proportional to the current population size. On the other hand, radioactive decay involves quantities decreasing over time. These models find applications in various fields, from biology (population growth) to physics (radioactive decay) and even in everyday scenarios like carbonation loss in beverages.

Let’s take a look at some examples and their differential equations:

• Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature.
  • \(\frac{dT}{dt} = -k(T – T_{\text{env}})\),
    • where \(T\) is the temperature of the object, \(T_{\text{env}}\) is the ambient temperature, and \(k\) is a positive constant.

• For a first-order chemical reaction, the rate of reaction is proportional to the concentration of a reactant.
  • \(\frac{d[A]}{dt} = -k[A]\),
    • where \([A]\) is the concentration of the reactant and \(k\) is the rate constant.

• The growth of an investment with compound interest can be modeled by a differential equation.
  • \(\frac{dP}{dt} = rP\),
    • where \(P\) is the principal amount and \(r\) is the interest rate.

• The voltage across a capacitor in an RC circuit as it charges follows the equation:
  • \(\frac{dV}{dt} = \frac{1}{RC}(V_{\text{source}} – V)\),
    • where \(V\) is the voltage across the capacitor, \(V_{\text{source}}\) is the source voltage, \(R\) is the resistance, and \(C\) is the capacitance.

• For an isothermal process in a gas, the pressure change can be modeled as:
  • \(\frac{dP}{dV} = -\frac{nRT}{V^2}\),
    • where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of gas, \(R\) is the gas constant, and \(T\) is the temperature.