Question

A cone of radius 10 cm and height 16 cm is divided into two parts by a plane through the midpoint of its

Answer 1

Answer:

The ratio of their volumes (the top to bottom half) is 1/7

Step-by-step explanation:

The given parameters are;

The radius, r, of the cone = 10 cm

The height of the cone = 16 cm

The point at which the cone is divided = The midpoint of the its axis and parallel to its base

The ratio of the of the volume of the two parts is given as follows;

The volume, V of the entire cone = 1/3 × Base area (π·r²) × Height, h

Therefore;

V = 1/3 × π × 10² × 16 =1600·π/3 =  1675.52 cm³

The height at which the cone is divided = 16/2 = 8 cm

The height, h₁ of the uppermost divided cone = 8 cm

The ratio of the radius to the height of the cone = 10/16 = 5/8

The radius r₁, of the uppermost divided cone = The height of the uppermost divided cone × The ratio of the radius to the height of the cone

∴ r₁ = 8 × 5/8 = 5 cm

The volume, V₁, of the uppermost divided cone = 1/3 × (π·r₁²) × h₁

∴ V₁ = 1/3 × π × 5² × 8 = 200·π/3 = 209.44 cm³

Which gives;

The volume of the lower cone, V₂ =  V - V₁

V₂ =  1675.52 cm³ - 209.44 cm³ = 1400·π/3 = 1466.1 cm³

The ratio of their volumes V₁ : V₂ = V₁/V₂ = (200·π/3)/(1400·π/3) = 209.44/1466.1 = 1/7.