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How many spherical solid marbles, each having a radius of 0.2┬аcm, can be made from a solid sphere having a radius of 6┬аcm?
27000
18000
21000
9000
27000
Calculate the ratio of the radii R/r=6/0.2=30. Since the volume of a sphere is proportional to the cube of its radius, the number of marbles is simply 303=27000.
Radius of large solid sphere (R) = 6 cm, Radius of each small marble (r) = 0.2 cm
n=Volume┬аof┬аone┬аmarbleVolume┬аof┬аlarge┬аsphereтАЛ=34тАЛ╧Аr334тАЛ╧АR3тАЛ=(rRтАЛ)3
Calculate the ratio of the radii R/r=6/0.2=30. Since the volume of a sphere is proportional to the cube of its radius, the number of marbles is simply 303=27000.
Many students try to calculate the actual volumes (4/3╧Аr3) separately, which introduces unnecessary calculations with ╧А and fractions, increasing the chance of arithmetic error.
Identify volumes
The number of marbles n is obtained by dividing the volume of the large sphere by the volume of one small marble.
n=34тАЛ╧Аr334тАЛ╧АR3тАЛ
Simplify ratio
Cancel common factors 34тАЛ╧А to get the ratio of radii cubed.
n=(rRтАЛ)3
Calculate result
Substitute R=6 and r=0.2. Dividing 6 by 0.2 gives 30. Finally, 30 cubed equals 27000.
n=(0.26тАЛ)3=(30)3=27000
A is correct because calculating the ratio of volumes (6/0.2)3 yields exactly 27000.
This type of scaling problem appears frequently in geometry; remember that volume scales by the cube of the scale factor (k3), whereas area scales by the square (k2).
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