RC settling time (repaired for solvability): A first-order RC output is considered “at maximum value” for practical purposes when it has reached about 99.3% of its final level (i.e., after 5 time constants). If the circuit’s time constant is 243.9 µs, how long does it take to reach this practical maximum?

Introduction / Context:
Because an exponential response never reaches its final value in strictly finite time, engineers adopt practical settling criteria such as 63.2% (1τ), 95% (≈3τ), 99.3% (≈5τ), or tighter depending on accuracy needs. This question uses the widely accepted 5τ rule of thumb to define when the output is “at maximum value” for typical design work.

Given Data / Assumptions:

• First-order RC circuit with time constant τ = 243.9 µs.
• “Maximum value” is defined here as 99.3% of final (≈ 5τ).
• Start from 0 V step to a new DC level; linear components.

Concept / Approach:
The exponential approach to the final value follows V(t) = V_final * (1 − e^(−t/τ)). At t = 5τ, e^(−5) ≈ 0.0067, so V(5τ) ≈ 0.993 * V_final. Multiply the given τ by 5 to find the practical settling time.

Step-by-Step Solution:

Verification / Alternative check:
For a tighter 7τ criterion (≈ 99.9%), time would be ≈ 1.71 ms; for a looser 3τ criterion (≈ 95%), time would be ≈ 0.732 ms. These bracket the 1.22 ms result logically.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing “time constant” with “settling time”; assuming “steady state” occurs at a fixed calendar time regardless of τ; mixing µs and ms units.

Final Answer:
1.22 ms