the height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volume of the two parts.
Answers
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
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