A right circular cone is cut by two planes parallel to the base and trisecting the altitude . What is the ratio of the volume of the three parts: top, middle bottom respectively?

1 : 7 : 19

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1 : 2 : 3

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1 : 8 : 27

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1 : 7 : 18

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Solution

The correct option is A $1 : 7 : 19$
$Δ A B C \sim Δ A D E \sim Δ A F G$
$∴ h_{1} : h_{2} : h_{3} = r_{1} : r_{2} : r_{3}$
$h_{1} : h_{2} : h_{3} = 1 : 2 : 3$
$∴ r_{1} : r_{2} : r_{3} = 1 : 2 : 3$
Volume of smaller cone $= \frac{1}{3} π (r_{1})^{2} (h_{1})$
Required ratio
$= [(1)^{2} \times 1] : [(2)^{2} \times 2 - (1)^{2} \times 1] : [3^{2} \times 3 - (2)^{2} \times 2]$
$= 1 : 7 : 19$

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Similar questions

A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19.

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio fo the volume of the smaller cone to the whole cone is

(a) 1 : 2 (b) 1 : 4 (c) 1 : 6 (d) 1 : 8

Q. The height of a right circular cone is trisected by two places drawn parallel to the base. Show that the volumes of the three portions starting from the top are in the ratio

1 : 7 : 19

Q. Choose the correct answer of the following question:

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

(a) 1 : 2 (b) 1 : 4 (c) 1 : 6 (d) 1 : 8 [CBSE 2012]

Q. If the right circular cone is separated into solids of volumes

v_{1}, v_{2}, v_{3}

by two plane parallel to the base and trisect the altitude

v_{1} : v_{2} : v_{3}

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A right circular cone is cut by two planes parallel to the base and trisecting the altitude . What is the ratio of the volume of the three parts: top, middle bottom respectively?

The correct option is A 1:7:19ΔABC∼ΔADE∼ΔAFG∴h1:h2:h3=r1:r2:r3h1:h2:h3=1:2:3∴r1:r2:r3=1:2:3 Volume of smaller cone =13π(r1)2(h1)Required ratio=[(1)2×1]:[(2)2×2−(1)2×1]:[32×3−(2)2×2]=1:7:19