A $0.5 \: m \times 0.5 \: m$ square concrete pile is to be driven in a homogeneous clayey soil having undrained shear strength $c_u=50 \: kPa$ and unit weight, $\gamma = 18.0 \: kN/m^3$. The design capacity of the pile $500$ kN. The adhesion factor $\alpha$ is given as $0.75$. The length of the pile required for the above design load with a factor of safety of $2.0$ is

- $5.2$ m
- $5.8$ m
- $11.8$ m
- $12.5$ m