the height of a cone is 30cm. a small cone is cut off at the top by a plane parallel to its base. if its volume be 1/27 of the volume of the given cone at what height above the base is the section made
Answers
Answer:
Let the Height and Radius of the Big cone be H cm and R cm
And of small cone be h cm and r cm
Let the volume of the big cone be V and of small cone be v
The cone is cut from the big cone then , they both cones would be similar to each other and according to the property of the similarity ,
H/R = h/r
⇒ h= Hr/R
⇒ h = 30 r/R ...............................eqn 1
Volume of the cone = 1/3 πr²h
v/V = r²h/R²H = 1/27
⇒ r²h = 30 R²/27
⇒ h=10 R²/9 r² ........................... eqn 2
On substituting the value of the h from eqn 1 in eqn 2 we get : -
30r/R = 10R²/9r²
⇒ r³/R³ = 1/27
⇒ r/R = 1/3
Substituting this ratio in eqn 1 we get :-
h = 30 × 1/ 3
⇒ h = 10 cm Ans.
Hence, Height of the small(cut) cone is 10 cm Ans.
And the Height of the Frustum is 30-10 = 20 cm Ans.
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