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A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height and radius of the cone so formed?
- a)4 : 3
- b)2 : 3
- c)3: 4
- d)1: 2
- e)None of these
Correct answer is option 'E'. Can you explain this answer?
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A solid sphere is melted and recast into a right circular cone with a ...
Volume of solid sphere = volume of cone
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A solid sphere is melted and recast into a right circular cone with a ...
To solve this problem, let's consider the properties of both the sphere and the cone.
Let the radius of the sphere be r. The volume of the sphere is given by V_sphere = (4/3)πr^3.
When the solid sphere is melted and recast into a cone, the volume of the cone will be equal to the volume of the sphere. The volume of a cone is given by V_cone = (1/3)πr^2h, where h is the height of the cone.
Setting the two volumes equal, we have:
(4/3)πr^3 = (1/3)πr^2h
Canceling out π and dividing both sides by (1/3)r^2, we get:
4r = h
So, the height of the cone is 4 times the radius of the cone.
Hence, the ratio of the height to the radius of the cone is 4:1.
Therefore, the correct answer is option E) None of these.
Let the radius of the sphere be r. The volume of the sphere is given by V_sphere = (4/3)πr^3.
When the solid sphere is melted and recast into a cone, the volume of the cone will be equal to the volume of the sphere. The volume of a cone is given by V_cone = (1/3)πr^2h, where h is the height of the cone.
Setting the two volumes equal, we have:
(4/3)πr^3 = (1/3)πr^2h
Canceling out π and dividing both sides by (1/3)r^2, we get:
4r = h
So, the height of the cone is 4 times the radius of the cone.
Hence, the ratio of the height to the radius of the cone is 4:1.
Therefore, the correct answer is option E) None of these.
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