# If the height of a right circular cone is 20 cm .A small cone is cut off at the top by a plane parallel to the base .If its Volume be 1/8 of Volume of the given cone,Find the h of small cone

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by NieuSidhu816

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by NieuSidhu816

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H = 20 cm = height of the right circular cone.

R cm = radius of the base of the cone

V = Volume of the Cone = 1/3 * π R² * H

Let the radius of the base of the small cone = r cm

h = height of the small cone.

v = volume of small cone = 1/3 π * r² * h

From the similar triangles principles,

r / h = R / H

r = R h / H

given V = 8 v

=> 1/3 π R² H = 8 * 1/3 π r² h

=> R² H = 8 * r² h

=> R² H = 8 * (R² h² / H²) * h

=> H³ = 8 h³

=> h = H/2

=> h = 20 cm / 2 = 10 cm

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another way:

When a small cone is cut off from the top of the cone, the ratio of the radii of the bases is equal to the ratio of the heights.

R / r = H / h = k (let us say)

R = k r and H = k h

Ratio of volumes = (π/3 R² H) / (π/3 r² h) = 8 given

=> ( k² r² k h ) / ( r² h ) = 8

=> k³ = 8

k = ∛8 = 2

=> H = 2 h and R = 2 r

Hence, the height of the small cone = H/2 = 10 cm.

R cm = radius of the base of the cone

V = Volume of the Cone = 1/3 * π R² * H

Let the radius of the base of the small cone = r cm

h = height of the small cone.

v = volume of small cone = 1/3 π * r² * h

From the similar triangles principles,

r / h = R / H

r = R h / H

given V = 8 v

=> 1/3 π R² H = 8 * 1/3 π r² h

=> R² H = 8 * r² h

=> R² H = 8 * (R² h² / H²) * h

=> H³ = 8 h³

=> h = H/2

=> h = 20 cm / 2 = 10 cm

============================

another way:

When a small cone is cut off from the top of the cone, the ratio of the radii of the bases is equal to the ratio of the heights.

R / r = H / h = k (let us say)

R = k r and H = k h

Ratio of volumes = (π/3 R² H) / (π/3 r² h) = 8 given

=> ( k² r² k h ) / ( r² h ) = 8

=> k³ = 8

k = ∛8 = 2

=> H = 2 h and R = 2 r

Hence, the height of the small cone = H/2 = 10 cm.