Question

A cone is divided into two
parts by drawing a plane through the midpoint of its axis, parallel to its
base. Compare the volumes of two parts.

Answer 1

Concept we will be using:
(i) Volume of a cone= $\frac{1}{3} \pi r^2h$, where r=radius of the base of the cone and h is the height of the cone.

ii) Mid-point theorem: In a triangle, the line segment that joins the midpoints of the two sides of the triangle is parallel to the third side and half of it.
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Let the radius of the original cone be R and the height be H.
Then, the volume of the original cone = $\frac{1}{3} \pi R^2H$

The cone is divided into two equal parts by drawing a plane through the mid points of its axis and parallel to the base.

Then, height of the top part (Please refer to the image) will be half of the original height.

Then, the height of the small cone = $\frac{H}{2}$
And, the radius of the small cone =$\frac{R}{2}$

Volume of the small cone=$\frac{1}{3} \pi(\frac{R}{2})^2(\frac{H}{2}) =\frac{1}{3} \pi(\frac{R^2}{4}) (\frac{H}{2})=\frac{1}{24} \pi R^2H$

Therefore, volume of the frustum
=Volume of the original cone - Volume of the small cone
$= \frac{1}{3} \pi R^2H -\frac{1}{24} \pi R^2H$
$= \frac{8\pi R^2H-\pi R^2H}{24}$
$= \frac{7}{24} \pi R^2H$


Compare the volume of the two part:
Volume of the frustum : Volume of the small cone=$\frac{7}{24} \pi R^2H : \frac{1}{24} \pi R^2H$
 
Volume of the frustum : Volume of the small cone=7:1


Answer : Volume of  the frustum is 7 times the volume of the small cone.

Answer 2

Answer:

Concept we will be using:

(i) Volume of a cone= , where r=radius of the base of the cone and h is the height of the cone.

ii) Mid-point theorem: In a triangle, the line segment that joins the midpoints of the two sides of the triangle is parallel to the third side and half of it.

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Let the radius of the original cone be R and the height be H.

Then, the volume of the original cone =

The cone is divided into two equal parts by drawing a plane through the mid points of its axis and parallel to the base.

Then, height of the top part (Please refer to the image) will be half of the original height.

Then, the height of the small cone =

And, the radius of the small cone =

Volume of the small cone=

Therefore, volume of the frustum

=Volume of the original cone - Volume of the small cone

Compare the volume of the two part:

Volume of the frustum : Volume of the small cone=

 

Volume of the frustum : Volume of the small cone=7:1

Answer : Volume of  the frustum is 7 times the volume of the small cone.

Step-by-step explanation: