Question

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 of the volumes of
the two parts of the cone.
CBSE 2013​

Answer 1

Okay, so height is divided equally through mid point and we need to find it's ratio of volume.

→ Volume of Cone = ⅓ πr²h

→ Volume = ⅓ π 10² × h

→ V = ⅓ π × 100h

Now since the mid point of height being parallel to base is joining slant height hence slant height is also divided equally. [Due to theorem]

→ h' = h/2

By ratio of similarity we get,

→ h'/r' = h/r

→ h/2/r' = h/10

→ 1/r' = h/10 × 2/h

→ r' = 5

Hence, volume of smaller cone

→ ⅓ πr'²h' → ⅓ π 25h/2

Now, volume of remaining part

→ ⅓ π(100h - 25h/2)

→ ⅓ π 175h/2

Ratio of volumes

= (⅓ π 25h/2)/(⅓ π 175h/2)

→ 1/7 or 1:7

Answer is 1:7