Slope stability using stability number: For a c–φ soil slope, given cohesion c = 1.5 t/m^2, unit weight γ = 2.0 t/m^3, factor of safety F = 1.5, and stability number S_n = 0.05, compute the safe height H of the slope (in metres).

Introduction / Context:
The stability number method provides a quick estimate of the allowable or safe height of slopes in cohesive soils by relating shear strength, unit weight, and safety factor. It is widely used for preliminary design and checks before detailed numerical analysis.

Given Data / Assumptions:

• Cohesion c = 1.5 t/m^2.
• Unit weight γ = 2.0 t/m^3.
• Factor of safety F = 1.5.
• Stability number S_n = 0.05 (corresponding to slope geometry and friction angle from charts).

Concept / Approach:

The stability number is defined for safe design as S_n = c / (F * γ * H). Rearranging yields the slope height H when c, γ, F, and S_n are known.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: t/m^2 divided by (t/m^3) gives metres; inclusion of F and S_n keeps H positive and in a realistic range for cohesive slopes.

Why Other Options Are Wrong:

5 m and 8 m are too conservative for the given parameters; 12 m and 15 m exceed the safe height implied by S_n = 0.05.

Common Pitfalls:

Using S_n = c/(γH) without including F; mixing units (kN/m^2 vs t/m^2) without consistent conversion; misreading the chart-supplied S_n for the intended φ and slope angle.

Final Answer:

10 metres