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CAT Previous Year Paper | CAT Mensuration and Geometry Questions | Question 18

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Question 18 : A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to :

  1. 10
  2. 50
  3. 60
  4. 20

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Explanatory Answer

Method of solving this CAT Question from Mensuration and Geometry

Given that a solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8 : 27 : 27. We have to find the percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube.
Let us take the volume of the 5 cubes to be 1x3 , 1x3 , 8x3 , 27x3 , 27x3
Volume of the original cube = x3(1 + 1 + 8 + 27 + 27)
Volume of the original cube = 64x3
Sides of the original cube = ∛64x3 = 4x
Similarly, sides of the 5 smaller cubes = x , x , 2x , 3x , 3x .
Surface Area of a cube = 6a2
Surface Area of the original cube = (4x) × (4x) = 16x2
Surface area of the smaller cubes = x2 , x2 , 4x2 , 9x2 , 9x2
Sum of the surface areas of the smaller cubes = x2(1 + 1 + 4 + 9 + 9) = 24x2
Change in surface area = 24x2 – 16x2 = 8x2
%change = \\frac{8x^2}{16x^2}) × 100 = 50%
Hence the percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to 50%.

The question is "A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to :"

Hence, the answer is 50%

Choice B is the correct answer.

 

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