A cone is cut into 3 parts by two cuts that are parallel to the base such that the heights of three parts are equal. Find the ratio of volume of three parts.
Answer & explanation
Correct answer: option 4
The cuts that are parallel to the base;
Δ AO1B, Δ AO2C, Δ AO3D are similar triangles;
Therefore, ratio of height (h) = ratio of respective radius (r)
⇒ Volume of cone = \(\frac{1}{3}\) \(\pi \) r2 h = \(\frac{1}{3}\) \(\pi \) h3
Now,
Ratio of volume of three triangles Δ AO1B, Δ AO2C and Δ AO3D
= \(\frac{1}{3}\) \(\pi \) (AO1)3 : \(\frac{1}{3}\) \(\pi \) (AO2)3 : \(\frac{1}{3}\) \(\pi \) (AO3)3
= (AO1)3 : (AO2)3 : (AO3)3
= (1R)3 : (2R)3 : (3R)3
= 1 : 8 : 27
Ratio of volume of three parts = 1 : (8 -1) : (27 -8) = 1 : 7 : 19