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A solid sphere of radius 6 cm is melted and recast into smaller spheres of radius 2 cm. How many such spheres can be formed?
18
27
36
54
27
When a solid object is melted and recast into smaller objects of the same material, the total volume remains conserved. By equating the volume of the large solid sphere to the combined volume of n smaller spheres, we find that the number of spheres formed is 27. Thus, Option B is the correct choice.
When a solid object is melted and recast into smaller objects of the same material, the total volume remains conserved. By equating the volume of the large solid sphere to the combined volume of n smaller spheres, we find that the number of spheres formed is 27. Thus, Option B is the correct choice.
Imagine melting a large ball of clay into several small, equal-sized marbles; the total amount of clay stays the same, allowing you to count how many marbles you can make by dividing the total volume.
Volume stays true, divide old by new!
V=34тАЛ╧Аr3 тАФ Volume of a sphere of radius r
n=VsmallтАЛVlargeтАЛтАЛ=(rRтАЛ)3 тАФ Number of recast spheres
The principle of conservation of volume dictates that the total volume of matter remains constant during phase changes or physical reshaping when no material is lost. The volume of a sphere is given by the formula V=34тАЛ╧Аr3. Dividing the volume of the larger sphere by the volume of one smaller sphere yields the total number of smaller spheres that can be produced.
Volume conservation is the fundamental principle behind melting and recasting problems.
The radius of the large sphere is R=6┬аcm and the radius of the small sphere is r=2┬аcm.
The ratio of the radii is 26тАЛ=3, and cubing this ratio gives 33=27 spheres.
Allows efficient determination of material distribution during manufacturing and shaping processes.
Assumes zero material loss or wastage during melting and recasting.
Foundry work and metal casting
Glass blowing and reshaping plastic components
Constant: ╧АтЙИ722тАЛ or 3.14159
Option A (18), Option C (36), and Option D (54) are incorrect because they fail to correctly apply the cube of the radius ratio.
B is correct тАФ 27 smaller spheres of radius 2 cm can be formed from a solid sphere of radius 6 cm based on volume conservation.
For similar mensuration problems involving melting and recasting, always express the ratio of volumes in terms of the ratio of linear dimensions raised to the power of three for 3D shapes.
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