Question

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

Answer 1

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.

Answer 2

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