When do you use the formulas of the Coefficient of Variation? Can you set an example to justify your answer?

CV is best suited when the intrinsic distributions are exponential rather than Gaussian or other. When this is the case, the mean will equal the SD. Since it is a dimensionless ratio estimator, it is more suitable for processes that employ ratio scale
units (eg Kelvin). Queuing theory uses CV quite frequently since Markovian processes are typical. Markov process are typically exponential.

Another good example where CV is the best choice are predator-prey or birth-death type problems where solutions are usually exponentially distributed.