Question

The height of two circular cones are in the ratio 1:2 and the perimeters of their bases are in the ratio 3:4. Find the ratio of their volumes

Answer 1

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let h1 and h2 are the heights of two circular cones .
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given , ratio of heights = 1 : 2

h1 : h2 = 1 : 2

h1 / h2 = 1 / 2

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let r1 , r2 be the radius of the bases of circular cones.

given ratio of the perimeters of their bases

= 3 : 4

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2 π r1 : 2 π r2 = 3 : 4
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r1 : r2 = 3 : 4

r1 / r2 = 3 / 4

let,volume of first cone V1 =1/3 π r1^2 h1

volume of second coneV2=1 /3 πr2^2h2

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Find ratio of volumes :
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V1/ V2=(1 / 3 π r1^2 h1 ) / ( 1/3 π r2^2h2 )

=r1^2h1/ r2^2 h2 = ( r1/r2 )^2( h1 / h2)



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since , r1 / r2 = 3 / 4 , and h1 / h2 = 1 / 2
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( r1 / r2 )^2 ( h1 / h2 ) = ( 3/4)^2 (1 / 2 )

= 9 / 16 × 1 / 2 = 9 / 32 = 9 : 32

therefore ,

ratio of their volumes = 9 : 32

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Your Answer :V1 / V2 = 9 : 32
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Answer 2

Let the 2 circular cones be Cone A and Cone B.

Define x:

Height A : Height B = 1 : 2 (Given)

Let x be the constant ratio

Height A : Height B = 1x : 2x

Define y:

Perimeter A : Perimeter B = 3 : 4 (Given)

Let y be the constant ratio

Height A : Height B = 3y : 4y

Find the radius of cone A:

Perimeter = 2πr

3y = 2πr

r = 3y ÷ 2π = (3y/2π) units

Find the radius of cone B:

Perimeter = 2πr

4y = 2πr

r = 4y ÷ 2π = 4y/2π = (2y/π) units

Find the volume of Cone A:

Volume = 1/3 πr²h

Volume = 1/3 π (3y/2π)² (x)

Volume = 1/3π(9y²/4π²) (x)

Volume = ( 3xy²/4π ) unit³

Find the volume of Cone A:

Volume = 1/3 πr²h

Volume = 1/3 π (2y/π)² (2x)

Volume = 1/3 π (4y²/π²) (2x)

Volume = ( 8xy²/3π ) unit³

Find the ratio of the volume:

Volume A : Volume B = 3xy²/4π  : 8xy²/3π

Multiply the ratio by 12π:

Volume A : Volume B = 9xy²  : 32xy²

Divide the ratio by xy²:

Volume A : Volume B = 9  : 32

Answer: The ratio of the volume is 9 : 32