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
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Correct Answer: 1 : 2
Solution : Given: Radius of bigger sphere = 4 cm The radius of a smaller sphere = 2 cm Let the radius of the bigger sphere be $R$ and smaller be $r$. According to the question, Number of spheres formed = $\frac{\text{Volume of bigger sphere}}{\text{Volume of smaller spheres}}$ = $\frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3}$ = $\frac{R^3}{r^3}$ = $\frac{4^3}{2^3}$ = 8 Now, the surface area of the original sphere = $4\pi R^2$ = $4\pi (4)^2$ = $64\pi$ And sum of surface area of small spheres = $8 × 4\pi (2)^2$ = $128\pi$ The required ratio = $64\pi : 128\pi= 1 : 2$ Hence, the correct answer is 1 : 2.
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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 : A thousand solid metallic spheres of 6 cm diameter each are melted and recast into a new solid sphere. The diameter of the new sphere (in cm) is:
Option 1: 30
Option 2: 90
Option 3: 45
Option 4: 60
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
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
Question : A solid metallic sphere of radius 12 cm is melted and recast into a cone having a diameter of the base of 12 cm. What is the height of the cone?
Option 1: 258 cm
Option 2: 192 cm
Option 3: 166 cm
Option 4: 224 cm
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