The cube has six identical square faces, each of which is 30 by 30. Therefore, the surface area of the cube is $6(30^{2})=5400$. The rectangular solid has two faces that are 20 by 30, two faces that are 20 by $L$, and two faces that are 30 by $L$. Thus, the surface area of the rectangular solid is $2(20)(30)+2(20L)+2(30L)=100L+1200$. Since the surface areas of the two solids are equal, then $100L+1200=5400$ or $100L=4200$, and so $L=42$.