A right circular cone having height of 30 cm is cut by two parallel planes at heights 10 cm and 20 cm from the base. What is the ratio of the volumes of the three parts (from top to bottom)?
1 : 7 : 19
For a cone, similar (smaller) cones formed by horizontal cuts are similar to the original. If the small cone has height \(h\) (measured from the apex), its volume scales as \(h^3\).
The apex is at the top. Heights from the apex of the three cut levels are:
• Top (small cone) ends at height 10 from apex (= 30 − 20 from base).
• Middle cut ends at height 20 from apex (= 30 − 10 from base).
• Full cone has height 30 from apex.
Volumes of the three full cones from the apex (in arbitrary units \(\propto h^3\)):
\(V(10) = 10^3 = 1000,\quad V(20) = 20^3 = 8000,\quad V(30) = 30^3 = 27000\)
Volumes of the three pieces:
Top piece (small cone) \(= V(10) = 1000\)
Middle frustum \(= V(20) - V(10) = 8000 - 1000 = 7000\)
Bottom frustum \(= V(30) - V(20) = 27000 - 8000 = 19000\)
Ratio (top : middle : bottom):
\(1000 : 7000 : 19000 = 1 : 7 : 19\)
Hence the ratio is 1 : 7 : 19 — option (4).
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
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- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
