An external cylindrical mouthpiece of diameter 100 mm is fitted flush on the side of a large reservoir (so that approach velocity is negligible). The measured discharge is 0.05948 m^3/s. Assuming standard coefficients for an external mouthpiece running full, estimate the head of water above the mouthpiece centerline.

Introduction / Context:
Discharge through short tubes and mouthpieces differs from that through thin-plate orifices because the jet expands to run full, altering the discharge coefficient. For an external mouthpiece on a large tank, we estimate the head from the measured discharge using the appropriate coefficient of discharge.

Given Data / Assumptions:

• Diameter D = 0.10 m → area a = πD^2/4.
• Measured discharge Q = 0.05948 m^3/s.
• Large reservoir: velocity of approach neglected.
• External cylindrical mouthpiece running full; coefficient of discharge C_d ≈ 0.82 (typical).

Concept / Approach:
For a mouthpiece: Q = C_d * a * sqrt(2 * g * H). Rearranging gives H = (Q / (C_d * a))^2 / (2 * g). We insert the measured Q and standard C_d to compute the head over the mouthpiece centerline.

Step-by-Step Calculation:
a = π * (0.10)^2 / 4 ≈ 7.854 × 10^-3 m^2.Velocity V = Q / a ≈ 0.05948 / 7.854e-3 ≈ 7.58 m/s.Ideal jet speed sqrt(2gH) = V / C_d ≈ 7.58 / 0.82 ≈ 9.24 m/s.Compute head H = (9.24^2) / (2 * 9.81) ≈ 85.4 / 19.62 ≈ 4.35 m.Closest listed value ≈ 4.0 m.

Verification / Alternative Check:
Using C_d in the 0.80–0.83 range gives H between roughly 4.2 and 4.5 m; 4.0 m is the closest discrete option and consistent with typical textbook rounding and experimental variability.

Why Other Options Are Wrong:
2.0–3.0 m: too small for the measured discharge and area with realistic coefficients.6.0 m: overestimates head unless coefficients are unreasonably low.

Common Pitfalls:

• Using orifice C_d (≈0.62) instead of mouthpiece C_d (≈0.82), leading to an erroneously large head.
• Ignoring approach velocity corrections when the supply conduit is not a large tank.
• Confusing internal/re-entrant mouthpieces with external cylindrical mouthpieces.

Final Answer:
4.0 m