Question

The hieght of a cone is 30cm. a small cone is cut off at the top by a plane parallel to the base. if its volume be 1/27 of the volume of the given cone at what height above the base the section has been made

answer is

Answer 1

Hello !
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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Answer 2

Let the height and radius of original(big) cone be H and R 
Let the height and radius of cut off (small) cone be h and r 
from similar triangles, we know 
H / R = h / r 
h = H r / R 
since H = 30 
h = 30 (r /R) --------(1) 
Volume of big cone, V = (1/3) π (R²) H 
volume of small cone,v = (1/3) π (r^2) h 
V / v = (R²) H / (r²) h = 27 (since volume of small cone is 1/27 of big cone) 
(R²) H = 27(r²) h 
30 (R²) = 27(r²)h 
h = 30(R²) / 27 (r²) 
h = (30 /27)(R /r)²---------(2) 
equating (1) and (2) 
30 (r /R) = (30 /27) (R/r)^2 
(r /R)³= 1 / 27 
r / R = 1/3 
substituting this in (1) h = 30 (r /R) 
h = 30 (1/3) = 10 cm 
The section is made 20 cm (30 -10) above the base.