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A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Choose value of π as found appropriate.)

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Short Answer Type

442.
A cistern, internally measuring 150 cm × 120 cm × 110 cm, has 129600 cm³ of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each brick being 22.5 cm × 7.5 cm × 6.5 cm?

Volume of water in the cistern
= 129600 cm³Let l, b and h are the length, breadth and height of the cistern. Then
l = 150 cm, b = 120 cm and h = 110 cm
Now, Volume of cistern = l x b x h
= 150 x 120 x 110 = 1980000 cm³∴ Volume of cistern to be filled
= (1980000 – 129600) cm³= 1850400 cm³Volume of one brick
= (22.5 x 7.5 x 6.5) cm³= 1096.875 cm³Let the total number of bricks be x.
then, water absored by x bricks

$equals space open parentheses straight x over 17 cross times 1096.875 close parentheses space cm cubed$

$therefore$ Volume of the water left in the cistem

$equals space open parentheses 129600 minus straight x over 17 straight x 1096.875 close parentheses space cm cubed$

Since, the cistern is filled upto the brim. Therefore,
Volume of the cistern
= Volume of the water left in the cistern + volume of the bricks

$rightwards double arrow space 1980000 space equals space open parentheses 129600 minus straight x over 17 straight x 1096.875 close parentheses space plus space straight x space cross times space 1096.875 rightwards double arrow space straight x space equals space 1792.410$

Hence, total no of bricks = 1792 (approx).

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443.

In one fortnight of a given month, there was a rainfall of 10 cm in a river valley. If the area of the valley is 7280 km2, show that the total rainfall was approximately equivalent to the addition to the normal water of three rivers each 1072 km long, 75 m wide and 3 m deep.

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444.

An oil funnel made of tin sheet consists of a 10 cm long cylindrical portion attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, find the area of the tin sheet required to make the funnel (see the given Fig.

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Long Answer Type

445.

Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

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Short Answer Type

446.

Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

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447. The radius and height of a cylinder are in the ratio 5:7 and its volume is 550 cm³. Find its radius.

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448. The curved surface area of height circular cone is 12320 cm². If the radius of the base is 56 cm, find its height.

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449. A conical flask is full of mater. The flask was base m radius r and height n. the mater is poored into a cylindrical flask of base radius mr. find the height of water in the cylindrical flask.

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450. A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?

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