Impedance claim sanity check — “The impedance is 9684 Ω”: Given a series RLC with R = 3.0 kΩ, X_L = 7.5 kΩ, and X_C = 5.5 kΩ at the operating frequency, evaluate the claim that the circuit impedance magnitude is 9684 Ω.

Introduction / Context:Numerical checks prevent order-of-magnitude mistakes in AC design. Series RLC impedance magnitude combines resistance and net reactance vectorially, not arithmetically. This item asks you to confirm or refute a specific impedance claim using given R, X_L, and X_C values.

Given Data / Assumptions:

• Series RLC with R = 3.0 kΩ.
• Inductive reactance X_L = 7.5 kΩ.
• Capacitive reactance X_C = 5.5 kΩ.
• Linear, sinusoidal steady state.

Concept / Approach:For series RLC, Z = R + j(X_L − X_C). The magnitude is |Z| = √(R^2 + (X_L − X_C)^2). Reactances must be netted before combining with R. Plug in values carefully and keep units consistent (ohms).

Step-by-Step Solution:

Verification / Alternative check:Phasor diagram shows a modest inductive reactance remaining; the impedance should be a few kilohms, not almost 10 kΩ. A quick calculator or spreadsheet confirms 3.606 kΩ.

Why Other Options Are Wrong:

Common Pitfalls:Adding R and reactances directly; forgetting to difference X_L and X_C before forming the magnitude; dropping kilo- prefixes during arithmetic.

Final Answer:Incorrect — |Z| ≈ 3606 Ω, not 9684 Ω