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 total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
