Question : A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes?
Option 1: 4 : 7
Option 2: 7 : 8
Option 3: 7 : 12
Option 4: 2 : 3
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Correct Answer: 7 : 8
Solution : Given, A solid metallic cuboid of dimensions 18 cm × 36 cm × 72 cm where l = 18 cm, b = 36 cm and h = 72 cm It is melted and recast into 8 cubes of the same volume. We know that, The volume of cuboid = lbh The total surface area of the cuboid = 2(lb + bh + hl) Where l = length, b = breadth and h = height The volume of cube = a3 The lateral surface area of cube = 4 × a2 Where a = side of the cube According to the question, lbh = 8 × a3 ⇒ 18 × 36 × 72 = 8 × a3 ⇒ a3 = 18 × 36 × 9 ⇒ a = $\sqrt[3]{9 × 2 × 9 × 2 × 2 × 9}$ ⇒ a = 18 cm Also, 2(lb + bh + hl) : 8 × 4 × a2 ⇒ 2(18 × 36 + 36 × 72 + 72 × 18) : 8 × 4 × (18)2 ⇒ 2 × 18 × 36(1 + 4 + 2) : 8 × 4 × 18 × 18 ⇒ 36 × 36 × 7 : 32 × 18 × 18 ⇒ 7 : 8 Hence, the correct answer is 7 : 8.
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Question : A solid metallic cuboid of dimensions 12 cm × 54 cm × 72 cm is melted and converted into 8 cubes of the same size. What is the sum of the lateral surface areas (in cm2) of 2 such cubes?
Option 1: 2268
Option 2: 1944
Option 3: 2592
Option 4: 3888
Question : A solid metallic sphere of radius 4 cm is melted and recast into spheres of 2 cm each. What is the ratio of the surface area of the original sphere to the sum of the surface areas of the spheres, so formed?
Option 1: 2 : 1
Option 2: 2 : 3
Option 3: 1 : 2
Option 4: 1 : 4
Question : A metallic cube has sides of length 12 cm. It is melted and recast into three small cubes. Of these cubes, two have sides measuring 6 cm and 8 cm, respectively. The length of each side of the third cube is:
Option 1: 7 cm
Option 2: 9 cm
Option 3: 10 cm
Option 4: 5 cm
Question : Three solid metallic spheres of radii 1 cm, 6 cm, and 8 cm, respectively, are melted and recast into a single solid sphere. The radius of the new sphere formed is:
Option 1: 9.0 cm
Option 2: 5.9 cm
Option 3: 7.7 cm
Option 4: 8.5 cm
Question : The radius of a large solid sphere is 14 cm. It is melted to form 8 equal small solid spheres. What is the sum of the total surface areas of all 8 small solid spheres? (use $\pi=\frac{22}{7}$)
Option 1: 3648 cm2
Option 2: 4928 cm2
Option 3: 4244 cm2
Option 4: 4158 cm2
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