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Answer 1

Question :-

A right cylindrical container of radius 6 cm and height 15 cm is full of ice cream, which has to be distributed to 10 children in equal cones having hemispherical shape on the top. If the height of the conical portion is four times its base radius, find radius of the ice cream cone.

First let us find the volume of icecream stored in the cylinder.

Volume of cylinder = πr²h

Given, r = 6 and h = 15 cm

→ volume = π(6)²(15)

→ volume = 540π

Now, this icecream is to be divided among 10 children in cones having hemispherical top with equal volumes

→ volume of ice cream in one cone = 540π/10

→ volume of icecream in on cone = 54π

let radius of the base be r, then the height of cone would be 4r according to the condition.

Now, volume of cone = 1/3 × πr²h

But here, hemispherical top will also have some volume.

so, total volume of icecream stored in cone

= volume of cone + volume of hemisphere

→1/3 × πr²h + 2/3 × πr³

(Here, radius will be the same as the hemisphere is made on the top of cone)

And, volume of icecream in one cone is 54π as we have shown above

→1/3 × πr²h + 2/3 × πr³ = 54π

→ 1/3 × πr²(4r) + 2/3 × πr³ = 54π

(since, h = 4r)

→ 1/3 × 4πr³ + 2/3 × πr³ = 54π

→ πr³(4/3 + 2/3) = 54π

→ πr³(6/3) = 54π

→ πr³(2) = 54π

→ 2πr³ = 54π

→ r³ = 54π/2π

→ r³ = 27

→ r = $\sqrt[3]{27}$

→ r = 3 cm

Hence, the base radius is of 3 cm

Answer :- 3 cm

Answer 2

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