Exponential models and numerical equations

In \(\mathrm{N}(t)=\mathrm{N}_0e^{kt}\), \(\mathrm{N}_0\) is the initial quantity and \(k\) is the continuous growth rate. A positive \(k\) means growth and a negative \(k\) means decay. Use the time units in the model.

Example

For \(\mathrm{N}(t)=600e^{-0.2t}\), \(\mathrm{N}(0)=600\). The half-life satisfies \(e^{-0.2t}=\dfrac12\), giving \(t=\dfrac{\ln2}{0.2}\approx3.47\) time units.

For a threshold depending on time, such as \(600e^{-0.2t}=100+20t\), plot both sides on \(t\ge0\) and locate their intersection numerically. Check that the result lies in the practical domain.

A model assumes its relationship continues over the stated interval. Explain whether extrapolation, constant rates and unlimited resources are reasonable before interpreting a prediction.