Derive $({}_1-{}_3{)}_f={}_{3f}[(\frac{{}_1^{\text{'}}}{{}_3}{)}_{\text{m}\text{a}\text{x}}-1].$ Start by using: $(\frac{{}_1^{\text{'}}}{{}_3^{\text{'}}}{)}_{\text{m}\text{a}\text{x}}=\frac{1+sin{}^{\text{'}}}{1-sin{}^{\text{'}}}=ta{n}^2(45+\frac{{}^{\text{'}}}{2}),$

and then draw the Mohr circle and Mohr$-$coulomb envelope at failure for sand.