Lesson 5 of 11
The hollow that trades
Show on a big screenA grinding stone has a bowl cut out of its top.
Step 1 of 4
- A granite block, seen from the side.
- A bowl-shaped hollow is cut from the top.
- The curved bowl is new skin, not a hole.
- The flat mouth is gone; the bowl remains.
Remove the mouth, add the bowl
The block is 14 by 14 by 10 units. A hemisphere of radius 7 is cut out of the top face. Here r = 7. The bowl's mouth, its opening, is a circle of radius 7. The book calls a hollow like this a cavity or a depression.
Surface area. The faces are 14 × 14 = 196 and 14 × 10 = 140.
- The whole block: 2(196 + 140 + 140) = 952.
- The mouth is cut away: πr² = 154.
- The bowl is new skin: 2πr² = 308.
- New surface = 952 − 154 + 308 = 1106.
Volume. The stone loses the volume of the bowl.
- Block: 14 × 14 × 10 = 1960.
- Bowl: (2/3)πr³ = 2156/3. Here r³ = 343, and (2/3) × 22/7 × 343 = 2156/3.
- To subtract: 1960 = 5880/3, and 5880 − 2156 = 3724.
- Volume = 1960 − 2156/3 = 3724/3, about 1241.33.
A hollow removes its flat mouth from the surface. It adds its curved bowl. A hemisphere bowl is twice the area of its mouth.
Cut the hollow and follow the surface.
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Try the lab to finish this lesson.