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There is a hemisphere hollow from inside in which there is a cone inscribed whose base is the diameter of the hemisphere. If the radius of the hemisphere is 6 cm, then find the ratio of the volume of hemisphere and volume of the remaining part of hemisphere when cone is dug out from it.

There is a hemisphere hollow from inside in which there is a cone inscribed whose base is the diameter of the hemisphere. If the radius of the hemisphere is 6 cm, then find the ratio of the volume of hemisphere and volume of the remaining part of hemisphere when cone is dug out from it. Correct Answer 2 : 1

Given:

Radius of the hemisphere = 6 cm

Diameter of base of hemisphere = Diameter of base of cone

Formula used:

Volume of hemisphere = 2πr3/3

Volume of cone = πr2h/3

As cone is inscribed in the hemisphere so, the height of the cone will be equal to the radius of the hemisphere

Calculation:

Radius of the base of hemisphere = 6 cm

⇒ Volume of hemisphere = 2π × 6 × 6 × 6/3 cm3 = 144π cm3

Now,

Radius of cone = radius of hemisphere = 6 cm

Height of cone = 6 cm

⇒ Volume of cone = π × 6 × 6 × 6/3 cm3 = 72π cm3

Now, Volume of the remaining part = Volume of hemisphere – volume of cone

⇒ Volume of the remaining part = 144π cm3 – 72π cm3 = 72π cm3

∴ Ratio of Volume of hemisphere and Volume of remaining part = 144π : 72π

⇒ 2 : 1

Short Trick:

Radius of both hemisphere and cone is equal and the height of the cone will also be equal to the radius of the hemisphere

So, h = r = 6 cm

Volume of the remaining part = 2πr3/3 – πr3/3         (as r = h)

⇒ Volume of the remaining part = πr3/3

⇒ π6 × 6 × 6/3 = 72π cm3

∴ Ratio of Volume of hemisphere and Volume of remaining part = 144π : 72π

⇒ 2 : 1

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