Final answer:
To determine the output response and time-based characteristics of a system, we often use methods such as the Laplace transform, and calculations for the time constant, rise time, and settling time based on system properties like resistance and capacitance.
Step-by-step explanation:
The question provided is about determining the output response c(t) of a given system, along with its time constant, rise time, and settling time. Since the system's exact description isn't provided, a general approach will be described:
- To find the output response c(t), one would typically perform a Laplace transform of the input and apply the system's transfer function to find the output in the Laplace domain. Then, an inverse Laplace transform is used to find c(t) in the time domain.
- The time constant is featured in systems that can be described by first-order differential equations, commonly found in RC (resistor-capacitor) circuits. The time constant (T) is calculated as T = RC.
- Rise time refers to the time a system takes to go from 10% to 90% of its final value during a step response.
- Settling time is the time required for the system output to settle within a certain percentage (usually 2% or 5%) of its final value.
Example: For an RC circuit with a resistance R = 1.00 x 10³ Ω and capacitance C = 8.00 µF, the time constant would be T = RC = (1.00 x 10³ Ω)(8.00 µF) = 8.00 ms.