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Question
If a hollow sphere of internal and external diameters 4 cm and 8 cm respectively melted into a cone of base diameter 8 cm, then find the height of the cone.
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Solution
In the given problem, we have a hollow sphere of given dimensions;
Internal diameter of the sphere (d) = 4 cm
External diameter of the sphere (D) = 8 cm
Now, the given sphere is molded into a cone,
Diameter of the base of cone (dc) = 8 cm
Now, the volume of hollow sphere is equal to the volume of the cone.
So, let the height of cone = h cm
Therefore, we get
Volume of cone = the volume of hollow sphere
`(1/3) pi ((d_c)/2)^2 h = (4/3) pi ((D/2)^3 -(d/2)^3)`
`(1/3) pi (8/2)^2 (h) = (4/3) pi ((8/2)^3 -(4/2)^3)`
`(1/3)pi (4)^2 (h) = (4/3) pi (64-8)`
Further, solving for h,
` h = ((4/3) pi (56))/((1/3) pi (16))`
`h = ((4)(56))/((16))`
h = 14 cm
So, height of the cone is 14 cm
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