Ratio of smaller cone and larger cone in a frustum
Answers
Answer:
Step-by-step
Let the height and radius of the given cone be H and R respectively.
The cone is divided into two parts by drawing a plane through the mid point of its
axis and parallel to the base.
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 r2 x h/2 = 1/6 πr2h
Volume of the cone OAD = 1/3 x π x R2 x h = 1/3 x π x 4r2 x h
The volume of the frustum = Volume of the cone OAD - Volume of the cone OCD
= (1/3 x π x 4r2 x h) – (1/3 x π x r2 x h/2)
= 7/6 πr2h
Ratio of the volume of the two parts = Volume of the cone OCD : volume of the frustum
= 1/6 πr2h : 7/6 πr2h
= 1 : 7