Sixty-four cubes of sides 2 cm each are combined to form a cube of side 8 cm. If four of the smaller cubes along the diagonal of a surface are removed from the surface of the large cube, which one of the following statements about the surface area of this solid object is true ?
The surface area of the large cube is 6 * (8 cm)^2 = 6 * 64 sq cm = 384 sq cm.
Four smaller cubes are removed along the diagonal of a surface (face) of the large cube. Consider one face as a 4×4 grid of the smaller cubes’ exposed faces. The diagonal consists of the 4 cubes at positions (0,0), (1,1), (2,2), (3,3) in this grid (assuming indexing from a corner 0,0).
These 4 small cubes correspond to positions in the 4x4x4 large cube grid. Let’s say the face is the top face (z=3, using 0-3 indexing). The cubes are (0,0,3), (1,1,3), (2,2,3), (3,3,3).
– Cubes (0,0,3) and (3,3,3) are corner cubes of the large cube assembly (3 faces originally exposed on the large cube’s surface).
– Cubes (1,1,3) and (2,2,3) are ‘face’ cubes of the large cube assembly (1 face originally exposed on the large cube’s surface).
When a small cube is removed from the surface of the large cube:
– Surface area is lost from the original surface of the large cube.
– New surface area is gained from the faces of the removed small cube that were previously internal.
– For a corner cube (like (0,0,3) or (3,3,3)): 3 faces (each 2×2=4 sq cm) were exposed on the large cube surface (total 12 sq cm). When removed, the 3 interior faces become exposed (total 12 sq cm). Net change in surface area = -12 + 12 = 0.
– For a face cube (like (1,1,3) or (2,2,3)): 1 face (4 sq cm) was exposed on the large cube surface. When removed, the 5 interior faces become exposed (total 5 * 4 = 20 sq cm). Net change in surface area = -4 + 20 = +16 sq cm.
The four removed cubes consist of 2 corner cubes and 2 face cubes of the large cube assembly.
Total change in surface area = (2 * Change from corner cube) + (2 * Change from face cube)
Total change = (2 * 0) + (2 * +16 sq cm) = 0 + 32 sq cm = +32 sq cm.
The surface area of the solid object after removing the cubes is the original surface area plus the net change.
New Surface Area = 384 sq cm + 32 sq cm = 416 sq cm.