Problem
A submerged stone displaces its own volume
A stone is put into a tank of water and the water level rises. We need the stone's volume.
Geometry
Your answer
How to solve
Strategy Draw a Diagram — The stone's own shape is unknowable and does not matter. What matters is that it takes up room the water used to have, so the water it pushed up is the same size as the stone -- and that pushed-up water is a neat rectangular slab.
1STEP 1
Find how far the water rose
Subtract the old height from the new one.
30 - 25 = 5
The water went up 5 cm.
6.RP.A.3Identify Subproblems2STEP 2
See that slab of water as the stone
No water was added, so the stone pushed that much water upward. The stone takes up exactly the room that slab of water now occupies.
The stone and the extra layer are the same size.
6.RP.A.3Visualize Spatial RelationshipsThe water that rose takes up as much room as the stone.
Why?
No water was added or spilled, so the water in the tank before and after is the very same water.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The stone and the water cannot be in the same place at once, so the water that used to fill the stone's space has to sit somewhere else, and the only place left is above the old level.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Measure that layer
The layer fills the whole floor of the tank and is 5 cm thick.
20 × 16 × 5 = 1600
The stone is 1600 cm³.
6.G.A.2Draw A DiagramAnswer
1600 cm³
Compare the two water volumes: 20 × 16 × 25 = 8000 before and 20 × 16 × 30 = 9600 after -- a difference of 1600 cm³, and no water was added.
Takeaway
A sunk object pushes up its own volume of water -- so measure the water instead of the object.
- Find how far the water rose
- See that slab of water as the stone
- Measure that layer