A two$-$tier column shown below is made from a material with modulus of $E=25$GPa. The top column

has length $L_1=2m,$ and cross$-$sectional area $A_1.$ The bottom column has length $L_2=1\mathrm{.}5m,$ and cross$-$

sectional area $A_2.$ The entire column is loaded by two forces, $P_1$ and $P_2$ as shown. Under the action of these

loads, the maximum permissible shortening of the column is $1\mathrm{.}0$ mm $.$

a$)$ If we require that $A_2=3A_1,$ what is the minimum permissible area $A_1?$

b$)$ If the areas $A_1$ and $A_2$ are such that the compressive stresses in both parts of the column are the same,

what is the minimum permissible area $A_1?$