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Surface Areas and Volumes

Short Answer Type

311.

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

When a semi-circular sheet is bent to form an open conical cup, the radius of the sheet becomes the slant height of the cup and the circumference of the sheet becomes the circumference of the base of the cone.

therefore space space space space space space space space space space space space l italic space equals 28 over 2 space cm space equals space 14 space cm

Let the base radius of the cup be r cm. Then,
2nr = π x 14

$rightwards double arrow space space space space space space 2.22 over 7. straight r equals 22 over 7.14 rightwards double arrow space space space space space space space space space space space space space space straight r equals 7 space cm$

Let the height of the cup be h cm. Then\

l² + r² + h²

$rightwards double arrow space space space space space space space space space space space space left parenthesis 14 right parenthesis squared equals left parenthesis 7 right parenthesis squared plus straight h squared rightwards double arrow space space space space space space space space space space space space space space straight h equals 7 square root of 3 space cm$

$therefore$ Capacity of the cup

$equals 1 third πr squared straight h equals 1 third.22 over 7. left parenthesis 7 right parenthesis squared.7 square root of 3 equals 1078 over 3 square root of 3 equals 1078 over 3 cross times 1.732 equals 622.36 space cm cubed$

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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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