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441.
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.)

In right triangle CAB :
BC² = AB² + AC²[Using Pythagoras theorem]
⇒ BC² = 3² + 4²⇒ BC² = 9 + 16 = 25
⇒ BC = 5 cm

In right triangle CAB :BC2 = AB2 + AC2[Using Pythagoras theorem]⇒

Now, in ΔAOB and ΔCAB :
∠AOB = ∠CAB (90°)
∠B = ∠B
[Common]
Therefore, by using A A similar condition

$increment AOB space minus space increment CAB OA over AC equals AB over CB rightwards double arrow space OA over 4 equals 3 over 5 rightwards double arrow OA equals 12 over 5 cm Similarly comma space space OB over AB equals AB over CB rightwards double arrow space space space space OB over 3 equals 3 over 5 rightwards double arrow OB equals 9 over 5 space cm therefore space space space space OC space equals space BC space minus space OB space space space space space space equals space 5 minus 9 over 5 equals fraction numerator 25 minus 9 over denominator 5 end fraction equals 16 over 5 space cm$

Now,
Volume of the double cone so formed

$equals space 1 third straight pi open parentheses 12 over 5 close parentheses squared straight x 16 over 5 plus 1 third straight pi open parentheses 12 over 5 close parentheses squared straight x 9 over 5 equals space 1 third straight pi open parentheses 12 over 5 close parentheses squared open square brackets 16 over 5 plus 9 over 5 close square brackets equals 1 third straight pi open parentheses 12 over 5 close parentheses squared open square brackets 25 over 4 close square brackets equals open parentheses 1 third straight pi cross times 12 over 5 cross times 12 over 5 cross times 12 over 5 close parentheses space cm cubed equals space open parentheses 3600 over 375 straight pi close parentheses space cm cubed equals 9.6 space straight pi space cm cubed equals space left parenthesis 9.6 space straight x space 3.14 right parenthesis space cm cubed equals space 30.14 space cm cubed$

and Surface area of the double cone

$equals space straight pi space straight x space 12 over 5 cross times 3 plus straight pi cross times 12 over 5 cross times 4 equals straight pi cross times 12 over 5 left square bracket 3 plus 4 right square bracket equals space straight pi cross times 12 over 5 cross times 7 equals 84 over 5 straight pi equals space 84 over 5 cross times 3.14 equals 52.75 space cm squared.$

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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?

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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.

450 Views

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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