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# (a) The ratio of the base radii of two right circular cones of the same height is 3 : 4. Find the ratio of their volumes. (b) The ratio of the heights of two right circular cones is 5 : 2 and that of their base radii is 2 : 5. Find the ratio of their volumes. (c) The height and the radius of the base of a right circular cone is half the corresponding height and radius of another bigger cone. Find: (i) the ratio of their volumes. (ii) the ratio of their lateral surface areas.

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An Important Question of M.L Aggarwal book of class 10 Based on Mensuration Chapter for ICSE BOARD.
The ratio of the base radii of two right circular cones of the same height is given.
Find the ratio of their volumes
This is the Question Number 13, Exercise 17.2 of M.L Aggarwal.

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1. (a) Let r1 and rbe the radius of the given cones and h be their height.

Ratio of radii, r1:r2 = 3:4

Volume of cone, V1 = (1/3)r12h

Volume of cone, V2 = (1/3)r22h

V1 /V1 = (1/3)r12h/ (1/3)r22h

= r12/ r22

= 32/42

= 9/16

Hence the ratio of the volumes is 9:16.

(b) Let h1 and hbe the heights of the given cones and r1 and rbe their radii.

Ratio of heights, h1:h2 = 5:2

Ratio of radii, r1:r2 = 2:5

Volume of cone, V1 = (1/3)r12h1

Volume of cone, V2 = (1/3)r22h2

V1 /V1 = (1/3)r12h1/ (1/3)r22h2

= r12 h1/ r22 h2

= 22×5/52×2

= 4×5/25×2

= 20/50 = 2/5

Hence the ratio of the volumes is 2:5.

(c) Let r be the radius of bigger cone. Then the radius of smaller cone is r/2.

Let h be the height of bigger cone. Then the height of smaller cone is h/2.

(i)Volume of bigger cone, V1 = (1/3)r2h

Volume of smaller cone, V2 = (1/3)(r/2)2 (h/2) = (1/3)r2h/8

Ratio of volume of smaller cone to bigger cone, V2/V1 = ( 1/3)r2 h/8÷ (1/3)r2h

= (1/24) r2 h ×(3/r2 h)

= 1/8

Hence the ratio of their volumes is 1:8.

(ii)slant height of bigger cone = √(h2+r2)

slant height of smaller cone = √((h/2)2+(r/2)2) = √(h2/4+r2/4) = ½ √(h2+r2)

Curved surface area of bigger cone, s1 = rl

r√(h2+r2)

Curved surface area of smaller cone, s2 = rl

×(r/2)× ½ √(h2+r2)

= ¼ r√(h2+r2)

ratio of curved surface area of smaller cone to bigger cone, s2/s1 = ¼ r√(h2+r2) ÷ r√(h2+r2)

= ¼ r√(h2+r2) ×1/(r√(h2+r2))

= 1/4

Hence the ratio of curved surface area of smaller cone to bigger cone is 1:4

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