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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?

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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.

Let R and r be respectively the radii of the bigger and smaller ends of the frustum, then

straight R space equals space 18 over 2 equals 9 space cm semicolon space space space straight r space equals space 8 over 2 equals 4 space cm

Let l and h be respectively the slant height and height of frustum, then
h = total height – height of cylindrical part
= 22 cm – 10 cm
= 12 cm

a n d space space space space space space space space l italic space italic space italic space equals space space italic space square root of straight h squared plus left parenthesis straight R minus straight r right parenthesis squared end root italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space equals space space space square root of left parenthesis 12 right parenthesis squared plus left parenthesis 9 minus 4 right parenthesis squared end root space space space space space space space space space space space space space space space space equals space space space space square root of 144 plus 25 end root space space space space space space space space space space space space space space space space equals space space space space square root of 169 space equals space 13 space cm

Now,
Curved surface area of frustum

$equals space rl space left parenthesis straight R space plus space straight r right parenthesis equals space 22 over 7 straight x space 13 space left parenthesis 9 plus 4 right parenthesis equals space space 22 over 7 straight x space 13 space straight x space 13 equals space 531.14 space cm squared$

Let r₁ and h₁ be respectively the radius and height of cylindrical part.
Then, r₁ = 4cm and h₁ = 10cm
Now,
Curved Surface area of the cylinder
= 2πr₁h _{1

$equals space open parentheses 2 space straight x space 22 over 7 straight x space 4 space straight x space 10 close parentheses space cm squared equals space 251.43 space cm squared$}

Hence,
Area of tin required
= Curved surface area of frustum + curved surface area of cylinder
= 531.14 + 251.43
= 782.57 cm².

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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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