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 ?<\/p>\n
[amp_mcq option1=”It is equal to the surface area of the large cube.” option2=”It is less than the surface area of the large cube.” option3=”It is more than the surface area of the large cube.” option4=”Insufficient data” correct=”option3″]<\/p>\n
When a small cube is removed from the surface of the large cube:
\n– Surface area is lost from the original surface of the large cube.
\n– New surface area is gained from the faces of the removed small cube that were previously internal.<\/p>\n
– 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.
\n– 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.<\/p>\n
The four removed cubes consist of 2 corner cubes and 2 face cubes of the large cube assembly.
\nTotal change in surface area = (2 * Change from corner cube) + (2 * Change from face cube)
\nTotal change = (2 * 0) + (2 * +16 sq cm) = 0 + 32 sq cm = +32 sq cm.<\/p>\n