According to Rankine’s formula, minimum depth of foundations, is A. $$\frac{{\text{P}}}{{\text{w}}} \times {\left( {\frac{{1 + \sin \varphi }}{{1 – \sin \varphi }}} \right)^2}$$ B. $$\frac{{\text{P}}}{{\text{w}}} \times {\left( {\frac{{1 – \sin \varphi }}{{1 + \sin \varphi }}} \right)^2}$$ C. $$\frac{{\text{P}}}{{2{\text{w}}}} \times {\left( {\frac{{1 – \sin \varphi }}{{1 + \sin \varphi }}} \right)^2}$$ D. $$\frac{{\text{P}}}{{\text{w}}} \times \frac{{1 + \sin \varphi }}{{1 – \sin \varphi }}$$

The correct answer is $\boxed{\frac{{\text{P}}}{{\text{w}}} \times {\left( {\frac{{1 – \sin \varphi }}{{1 + \sin \varphi }}} \right)^2}}$.

Rankine’s formula is used to calculate the minimum

$$d = \frac{{\text{P}}}{{\text{w}}} \times {\left( {\frac{{1 – \sin \varphi }}{{1 + \sin

where:

• $d$ is the minimum depth of foundation
• $P$ is the active earth pressure
• $w$ is the weight of the soil above the foundation
• $\varphi$ is the angle of internal friction of the soil

The formula takes into account the weight of the soil above the foundation, the active earth pressure, and the angle of internal friction of the soil. The minimum depth of foundation is calculated by dividing the active earth pressure by the weight of the soil above the foundation and multiplying by the square of the tangent of the angle of internal friction of the soil.

The other options are incorrect because they do not take into account the weight of the soil above the foundation.