A 60 Hz series RLC circuit has XL = 7.5 ohms, XC = 265.3 ohms, and R = 33 ohms. What is the magnitude of the total impedance Z?

Introduction:
This problem checks application of series RLC impedance calculation. In a series RLC, the total impedance combines resistance and net reactance vectorially, so you must compute the net reactance first and then the magnitude of Z using the Pythagorean relationship.

Given Data / Assumptions:

• Series RLC at 60 Hz.
• R = 33 ohms.
• XL = 7.5 ohms.
• XC = 265.3 ohms.

Concept / Approach:
The net reactance is X = XL − XC. The total impedance for a series circuit is Z = R + jX, whose magnitude is |Z| = sqrt(R^2 + X^2). Because XC is much larger than XL here, the circuit is net capacitive and the phase is negative; however, the question asks only for the magnitude of Z.

Step-by-Step Solution:

Verification / Alternative check:
Since |X| ≫ R, |Z| should be close to |X|. The answer 259.9 ohms is slightly larger than 257.8 ohms due to the added resistive component, which is consistent.

Why Other Options Are Wrong:

• 257.8 ohms: That is |X| alone, not the total |Z|.
• 290.8 ohms: Overestimates the vector sum; not supported by calculation.
• 1989.75 ohms: Numerically unrelated to any correct combination here.

Common Pitfalls:
Using arithmetic addition instead of vector magnitude, forgetting that X can be negative (capacitive), or substituting XL + XC instead of XL − XC for series networks.

Final Answer:
259.9 ohms.