Question : A solid brass sphere of radius 2.1 dm is converted into a right circular cylindrical rod of length 7 cm. The ratio of the total surface areas of the rod to the sphere is:
Option 1: 3 : 1
Option 2: 1 : 3
Option 3: 7 : 3
Option 4: 3 : 7
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Correct Answer: 7 : 3
Solution : Given: Radius of the sphere ($R$) = 2.1 dm = 21 cm Height of the rod = 7 cm Let the radius of the rod be $r$ cm. Now, The volume of the rod = volume of the sphere $⇒\pi r^2 h=\frac{4}{3}\pi R^3$ $⇒r^2\times 7 = \frac{4}{3}\times21\times21\times21$ $\therefore r = 42$ The ratio of the total surface area of the rod to the sphere $=2\pi r(h+r):4\pi R^2$ $=2 × 42 × (42 + 7): 4 × 21 × 21$ $=2 × 42 × 49: 4 × 21 × 21$ $=7: 3$ Hence, the correct answer is 7 : 3.
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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 solid copper sphere of radius 9 cm is hammered and moulded into a wire of radius 2 cm. What is the length of this wire?
Option 1: 224 cm
Option 2: 183 cm
Option 3: 198 cm
Option 4: 243 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
Question : The radius of the base of a hollow cone is 8 cm, and its height is 15 cm. A sphere of the largest radius is put inside the cone. What is the ratio of the radius of the base of a cone to the radius of a sphere?
Option 1: 5 : 3
Option 2: 4 : 1
Option 3: 2 : 1
Option 4: 7 : 3
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
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