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MENSURATION PROBLEM

PROBLEM: 
If two cubes with 64 cubic cms. each are combined to form into a cuboid, then what is the total surface area of such cuboid so formed?

SOLUTION:
 Given: Volume of cube = 64 cubic cms.
 It means, a.a.a = 64
                  a = cubeth root of 64
                  a = 4 cm.
Thus, side of each cube is 4 cms.
Now, if two cubes with side 4 cms each are combined into a cuboid, the length, l of cuboid would be 4+4=8 cms. It's height, h would be 4cms. and it's breadth, b would be 4cms.

TOTAL SURFACE AREA(TSA):

                                METHOD 1
Now, total surface area of cuboid = 2(lb+bh+lh) = 2(8*4 + 4*4 + 8*4) = 2(32+16+32) = 2(80) = 160sq.cms.
                                        METHOD 2
Since cuboid is formed by combining two cubes of 4cm. each, we find the sum of total surface areas of cubes. Total surface area of a cube = 6aa = 6*4*4 = 96 sq.cms.
Thus sum of total surface areas of two cubes = 2*96 = 192 sq.cm.
But this 192 will not become the TSA of cuboid because when 2 cubes are combined to form a cuboid, the place where 2 sides of cubes(one of each) are combined or attached will not form the surface of cuboid so formed. Thus, we need to subtract the area of such two sides that are included in the area calculated above (192 sq.cms). One side area is a.a (since it is a square) and so, the area of two sides is 2aa. 
Thus, TSA of cuboid = 192-(2a.a)= 192- (2*4*4) = 192-32 = 160 sq.cms.






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