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542. A tent is in the form of a cylinder surmounted by a conical top. If the height and diameter of the cylinderical part are 2.1 m and 4m respectively, and the height of the top is 2.8m, find the area of the canvas used for making the tent. Find the cost of the canvas used for making the tent. Find the cost of the canvas of the tent at the rate of Rs. 500 per m². Also, find the volume of air enclosed in the tent.

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543. A toy is in the form of a cone mounted on a hemisphere with same radius. The diameter of the base of the conical portion is 7.0 cm and the total height of the toy is 14.5 cm. Find

the volume of the toy

open square brackets Use space straight pi space equals space 22 over 7 close square brackets.

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544. An iron pillar has lower part in the form of a right circular cylinder and the upper part in the form of a right circular cone. The radius of the base of each of the cone and cylinder is 8 cm. The cylindrical part is 240 cm high and the conical part is 36 cm high. Find the weight of the pillar if 1 cm³ of iron weighs 7.5 grams.

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

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Short Answer Type

545. A bucket is made up of a metal sheet is in the form of a frustum of a cone of height 16cm with radii of its lower and upper ends at 8cm and 20cm respectively. Find the cost of the bucket, if the cost of metal sheet used is Rs. 15 per 100 cm².

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Long Answer Type

546. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be

1 over 27

of the volume of the given cone, at which height above the base is the section made?

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547. A hollow cone is cut by a plane parallel to the base and the upper portul is removed. If the curved surface of the remainder is $8 over 9$ of the curved surface of the whole cone, find the ratio of the line-segment into which the cones altitude is divided by the plane.

Let r and R be respectively the radii of upper portion of cone and whole cone, and h and H be their respective heights.
And let l and L be respectively the slant heights of upper portion of cone and whole cone.
It is given that :

$fraction numerator Curved space surface space of space remaining space portion over denominator Curved space surface space of space the space whole space cone end fraction equals 8 over 9 therefore space space fraction numerator table row cell table row cell table row cell Curved space surface space of space the space upper end cell row cell portion space of space the space cone end cell end table end cell end table end cell end table over denominator Curved space portion space of space the space whole space cone end fraction equals 1 over 9 rightwards double arrow space space space space space space πrl over πRL equals 1 over 9 rightwards double arrow space space space space space straight r over straight R straight x space straight l over straight L equals 1 over 9 rightwards double arrow space space straight h over straight H straight x straight h over straight G equals 1 over 9 space space open square brackets table row cell because space increment ABC space minus space increment ADE end cell row cell rightwards double arrow space straight r over straight R equals straight l over straight L equals straight h over 30 end cell end table close square brackets rightwards double arrow space space space open parentheses straight h over straight H close parentheses squared equals open parentheses 1 third close parentheses squared rightwards double arrow space space straight h over straight H equals 1 third rightwards double arrow space space space straight H space equals space 3 straight h therefore space Required space height space equals space fraction numerator Height space of space upper space portion over denominator Height space of space the space frustum end fraction rightwards double arrow space space space space fraction numerator straight h over denominator 3 straight h minus straight h end fraction equals fraction numerator straight h over denominator 2 straight h end fraction equals 1 half.$

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548. If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, find the capacity and total surface area of the bucket.

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549. A bucket is in the form of a frustum of a cone and holds 28.49 litres of milk. The radii of the top and bottom are 28 cm and 21 cm. Find the height of the bucket.

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Short Answer Type

550.

A bucket of height 8 cm made up of copper sheets is in the form of frustum of a right circular cone with radii of its lower ends as 3 cm and 9 cm respectively. Calculate
(i) the height of the cone of which the bucket is a part.
(ii) the volume of water which can be filled in the bucket.
(iii) the area of copper sheet required to make the bucket.

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