Optional extension: Cones, hollow solids and changing dimensions
Example 1
A hollow square column has uniform wall thickness 1cm, height 6cm and material volume 192cm3. The top and bottom are open. Find its outer side length x.
CuboidDiagram not to scaleShow worked answer
Answer: 9
The inner side is x−2.
Volume =6[x2−(x−2)2]=6(4x−4).
Solve for x=9.
Example 2
A cone of base radius 12cm and height 30cm is cut parallel to its base halfway up its height. Remove the smaller top cone. Find the remaining frustum volume in cm3. Use V=3πr2h and π=3.
Cone cut parallel to baseDiagram not to scaleShow worked answer
Answer: 3780
The removed cone has half the original linear dimensions: radius 6, height 15.
Subtract its volume from the full cone: 4320−540=3780.
Example 3
A hollow square column has uniform wall thickness 1cm, height 10cm and material volume 280cm3. The top and bottom are open. Find its outer side length x.
CuboidDiagram not to scaleShow worked answer
Answer: 8
The inner side is x−2.
Volume =10[x2−(x−2)2]=10(4x−4).
Solve for x=8.
Composite Solids II practice | Year 12 | Ascent Arcade