Final answer:
The transfer function can be approximated as a second-order system because the system's behavior is predominantly governed by the second-order dynamics, rather than the contribution from the faster pole.
Step-by-step explanation:
For the transfer function T(s) = 14.145/(s² + 0.842s + 2.829)(s + 5), considering the approximation as a second-order system is valid due to the relatively high pole at s = -5 compared to the two complex poles close to the imaginary axis from the second-order denominator (s² + 0.842s + 2.829). The significant difference in the magnitudes of the real parts of these poles means that the dynamics of the system will be predominantly governed by the response of the second-order component, and the contribution from the fast pole at s = -5 will decay quickly, making it negligible for the long-term response.
This is a common approximation in control theory when analyzing the dominant poles of a system. In essence, we are assuming that the system's behavior can be sufficiently described by the second-order dynamics due to the faster pole exerting minimal influence over the system's output after a short transient period. This simplification is beneficial for both analytical considerations and for understanding the system's behavior without having to deal with the complexities introduced by the higher-order terms.