the height of a cone is 30 cm. a small cone is cut off at the top by a plane parallel to the base.if its volume be 1/27th of the volume of the given cone, at what height above the base is section made?
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Assume that the small cone of height h is cut off from the top of this cone whose base is parallel to the big cone and the radius of the big cone be R and the radius of the small cone be r.
Given : Height = 30 cm
Volume of the big cone = 1/3πR²h
= 1/3πR²*30
= 10πR² cu cm
Given : volume of the small cone = 1/27th of the volume of the big cone.
Let us assume that V₁ be volume of big cone and V₂ be the volume of small cone.
V₂/V₁ = 1/27
⇒ (1/3πr²h/10πR²) = 1/27
⇒ (r²h/30R²) = 1/27
⇒ (r²/R²)² × (h/30) = 1/27 .......(1)
From the figure Δ ACD ~ Δ AOB (By AA similarity criterion)
⇒ r/R = h/30
Now, putting this value of (r/R = h/30) in equation (1), we get
(h/30)² × (h/30) = 1/27
⇒ (h/30)³ = 1/27
⇒ (h/30)³ = (1/3)³
h = 30/3
h = 10 cm
So, the height above the base where the section is made = 30 - 10 = 20 cm
Answer.
Assume that the small cone of height h is cut off from the top of this cone whose base is parallel to the big cone and the radius of the big cone be R and the radius of the small cone be r.
Given : Height = 30 cm
Volume of the big cone = 1/3πR²h
= 1/3πR²*30
= 10πR² cu cm
Given : volume of the small cone = 1/27th of the volume of the big cone.
Let us assume that V₁ be volume of big cone and V₂ be the volume of small cone.
V₂/V₁ = 1/27
⇒ (1/3πr²h/10πR²) = 1/27
⇒ (r²h/30R²) = 1/27
⇒ (r²/R²)² × (h/30) = 1/27 .......(1)
From the figure Δ ACD ~ Δ AOB (By AA similarity criterion)
⇒ r/R = h/30
Now, putting this value of (r/R = h/30) in equation (1), we get
(h/30)² × (h/30) = 1/27
⇒ (h/30)³ = 1/27
⇒ (h/30)³ = (1/3)³
h = 30/3
h = 10 cm
So, the height above the base where the section is made = 30 - 10 = 20 cm
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