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.
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