Question

The height of a cone is 10cm. The cone is divided into 2 parts using a plain parallel to it's base at the middle of it's height. Find the ratio of volume of 2 parts.

Answer 1

Let the height and radius of the given cone be H and R respectively.

Upper part is a smaller cone and the bottom part is the frustum of the cone.
⇒ OC = CA = h/2

Let the radius of smaller cone be r cm.

In ΔOCD and ΔOAB,
∠OCD = ∠OAB = 90°
∠COD = ∠AOB (common)
∴ ΔOCD ∼ ΔOAB (AA similarity)
⇒ OA/OC = AB/CD = OB/OD
⇒ h / h/2 = R/r
⇒ R = 2r

the radius and height of the cone OCD are r cm and h/2 cm

therefore the volume of the cone OCD = 1/3 x π x r x h/2 = 1/6 πr h

Volume of the cone OAD = 1/3 x π x R x h = 1/3 x π x 4r x h

The volume of the frustum = Volume of the cone OAD - Volume of the cone OCD
= (1/3 x π x 4r x h) – (1/3 x π x r x h/2)
= 7/6 πr h

Ratio of the volume of the two parts = Volume of the cone OCD : volume of the frustum
= 1/6 πr h : 7/6 πr h
= 1 : 7

Answer 2

Answer:

1:7

Step-by-step explanation:

If the cone is divided in two Equal parts from height ,

The Triangle AMD and ANC are similar ( by AAA similarity)

So AM/AN = MD/NC

Or 5/10 = r/R

or R =2r

The Volume of the upper small cone of height 5cm

=1/3(πr²h)

=πr²5/3

And Volume of the frustum BCDE

=Volume of whole cone of height 10 cm - volume of small cone height 5cm

=1/3[πR²10 -πr²5]

=1/3[5π(2R² -r²)]

=5π[2(2r)² -r²]/3

=5π[8r²-r²]/3

=5π(7r²)/3

=35πr²/3

So the ratio of Volumes

=(5πr²/3)/(35πr²/3)

=5/35

=1/7

=1:7