Question

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, the ratio of the volumes of upper and lower part is
A. 1 : 2
B. 2 : 1
C. 1 : 7
D. 1 : 8

Answer 1

Given : If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis.

 

SOLUTION :  

Let r & R be the radius of the lower part of the cone.

[Cut through the midpoint of its height]

From the figure,  

AB = h

AB’ = 2h

BC = r  

B'C = R

In ∆ABC & ∆AB’C’ ,

∠ABC = ∠AB’C’ (each 90°)

∠ACB = ∠AC’B’ (corresponding angles)

∆ABC ∼ ∆AB’C’ [By AA Similarity]

BC/B'C’ = AB/AB’

[Corresponding sides of a similar triangles are proportional]

r/R = h/2h

r/R = ½

R = 2r

Volume of the upper part (Smaller cone) = ⅓ πr²h

Volume of solid cone = ⅓ π R²2h

= ⅓ π (2r)² 2h = ⅓ π × 4r² × 2h = 8/3πr²h

Volume of solid cone = 8/3πr²h

Volume of lower part (frustum) = volume of solid cone - volume of Smaller cone   = 8/3πr²h - ⅓ πr²h = 7/3 πr²h

Volume of lower part (frustum) = 7/3 πr²h

Volume of the upper part (Smaller cone)/ Volume of lower part (frustum) =  

⅓ πr²h / 7/3 πr²h

= 1/7  

Hence, the ratio of volume of two parts of the cone is 1 : 7 .

Among the given options option (C) 1 : 7 is correct.

HOPE THIS ANSWER WILL HELP YOU……

 

Some more questions :  

A solid cone of base radius 10 cm is cut into two parts through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.

https://brainly.in/question/8927861

 

In the given fig from the top of a solid cone of height 12cm and base radius 6cm, a cone of height 4cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid(Use $(\pi=\frac{22}{7})$ √5=2.236)

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Answer 2

Answer:

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