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311.
If h, c, v are respectively the height, curved surface and the volume of a cone, prove that 3πvh³ – c²h² + 9v² = 0

Let the base radius and the height of the cone be r and h respectively.
Let the slant height of the cone be l.
Then,

c = $straight pi$ rl

$equals πr square root of straight r squared plus straight h squared end root straight v equals 1 third πr squared straight h therefore space space space space 3 πvh cubed minus straight c squared straight h squared plus 9 straight v squared space space space space space space space equals 3 straight pi open parentheses 1 third πr squared straight h close parentheses straight h cubed minus straight pi squared straight r squared left parenthesis straight r squared plus straight h squared right parenthesis straight h squared space plus space 9 open parentheses 1 third πr squared straight h close parentheses squared space space space space space space space space space space space space space space space$

= π²r²h² – π²r⁴h² – π²r⁴ + π²r⁴h⁴ = 0

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312.

A vessel is in the shape of a cone. Radius of the broader end is 2.1 cm and height is 20 cm. Find the volume of the vessel.

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313. The radius and height of a right circular cone are in the ratio of 5 : 12. If its volume is 314 cm³, find the slant height and radius of the cone, (

straight pi

= 3.14)

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314.

A conical tent is to accommodate 11 persons. Each person must have 4 square metre of the space on the ground and 20 cubic metres of air to breath. Find the height of the cone.

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315. A conical vessel whose internal radius is 5 cm and height 63 cm is full of oil. Find the the volume of the oil. If that oil is emptied into a cylindrical vessel with internal radius 10 cm, find the height of the oil in the cylindrical vessel.

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316. A semi-circular sheet of metal of diameter 28 cm is bent to form an open conical cup. Find the capacity of the cup.

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317. Find the volume of the largest right circular cone that can be placed in a cube of edge 7 cm.

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318. Bhavya has a piece of canvas whose area is 552 m². She uses it to make a conical tent with a base radius of 7 m. Assuming that all the stiching margins and the wastage incurred while cutting amounts to approximately 2 m², find the volume of the tent that can be made with it.

open parentheses Take space straight pi equals 22 over 7 close parentheses

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319. The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone.

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320. Monica has a piece of canvas whose area is 551 m². She uses it to have a conical tent made, with a base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 m², find the volume of the tent that can be made with it.

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