← A sunk object's volume is the water it pushes up · Surface Area and Volume of Solids

A sunk object's volume is the water it pushes up · 12 practice problems

6.G.A.2

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 1344 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 18 cm18\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 12 cm 16 cm 14 cm
Show solution
1 · Understandwhat's really being asked

A tank 16 cm by 14 cm holds water 12 cm deep. A stone goes in, fully under, and the water reaches 18 cm. We want the stone's volume.

Givens
  • The tank's base is 16 cm by 14 cm.
  • The water starts 12 cm deep and ends 18 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
18−12=618 - 12 = 6
The water climbed 6 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 16 by 14 base, 6 cm thick.
16×14×6=134416 \times 14 \times 6 = 1344
The stone is 1344 cm3.
Answer: 1344 cm³
4 · Reviewdoes it hold up?

The water before was 2688 cm3 and after is 4032 cm3; the difference, 1344 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 4032 minus 2688, the same 1344 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 2 easy answer: 1080 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 20 cm20\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 15 cm 18 cm 12 cm
Show solution
1 · Understandwhat's really being asked

A tank 18 cm by 12 cm holds water 15 cm deep. A stone goes in, fully under, and the water reaches 20 cm. We want the stone's volume.

Givens
  • The tank's base is 18 cm by 12 cm.
  • The water starts 15 cm deep and ends 20 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
20−15=520 - 15 = 5
The water climbed 5 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 18 by 12 base, 5 cm thick.
18×12×5=108018 \times 12 \times 5 = 1080
The stone is 1080 cm3.
Answer: 1080 cm³
4 · Reviewdoes it hold up?

The water before was 3240 cm3 and after is 4320 cm3; the difference, 1080 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 4320 minus 3240, the same 1080 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 3 easy answer: 2160 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 21 cm21\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 16 cm 24 cm 18 cm
Show solution
1 · Understandwhat's really being asked

A tank 24 cm by 18 cm holds water 16 cm deep. A stone goes in, fully under, and the water reaches 21 cm. We want the stone's volume.

Givens
  • The tank's base is 24 cm by 18 cm.
  • The water starts 16 cm deep and ends 21 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
21−16=521 - 16 = 5
The water climbed 5 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 24 by 18 base, 5 cm thick.
24×18×5=216024 \times 18 \times 5 = 2160
The stone is 2160 cm3.
Answer: 2160 cm³
4 · Reviewdoes it hold up?

The water before was 6912 cm3 and after is 9072 cm3; the difference, 2160 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 9072 minus 6912, the same 2160 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 4 easy answer: 1500 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 24 cm24\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 20 cm 25 cm 15 cm
Show solution
1 · Understandwhat's really being asked

A tank 25 cm by 15 cm holds water 20 cm deep. A stone goes in, fully under, and the water reaches 24 cm. We want the stone's volume.

Givens
  • The tank's base is 25 cm by 15 cm.
  • The water starts 20 cm deep and ends 24 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
24−20=424 - 20 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 25 by 15 base, 4 cm thick.
25×15×4=150025 \times 15 \times 4 = 1500
The stone is 1500 cm3.
Answer: 1500 cm³
4 · Reviewdoes it hold up?

The water before was 7500 cm3 and after is 9000 cm3; the difference, 1500 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 9000 minus 7500, the same 1500 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 5 medium answer: 2240 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 26 cm26\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 22 cm 28 cm 20 cm
Show solution
1 · Understandwhat's really being asked

A tank 28 cm by 20 cm holds water 22 cm deep. A stone goes in, fully under, and the water reaches 26 cm. We want the stone's volume.

Givens
  • The tank's base is 28 cm by 20 cm.
  • The water starts 22 cm deep and ends 26 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
26−22=426 - 22 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 28 by 20 base, 4 cm thick.
28×20×4=224028 \times 20 \times 4 = 2240
The stone is 2240 cm3.
Answer: 2240 cm³
4 · Reviewdoes it hold up?

The water before was 12320 cm3 and after is 14560 cm3; the difference, 2240 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 14560 minus 12320, the same 2240 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 6 medium answer: 2400 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 22 cm22\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 18 cm 30 cm 20 cm
Show solution
1 · Understandwhat's really being asked

A tank 30 cm by 20 cm holds water 18 cm deep. A stone goes in, fully under, and the water reaches 22 cm. We want the stone's volume.

Givens
  • The tank's base is 30 cm by 20 cm.
  • The water starts 18 cm deep and ends 22 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
22−18=422 - 18 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 30 by 20 base, 4 cm thick.
30×20×4=240030 \times 20 \times 4 = 2400
The stone is 2400 cm3.
Answer: 2400 cm³
4 · Reviewdoes it hold up?

The water before was 10800 cm3 and after is 13200 cm3; the difference, 2400 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 13200 minus 10800, the same 2400 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 7 medium answer: 1600 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 30 cm30\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 25 cm 20 cm 16 cm
Show solution
1 · Understandwhat's really being asked

A tank 20 cm by 16 cm holds water 25 cm deep. A stone goes in, fully under, and the water reaches 30 cm. We want the stone's volume.

Givens
  • The tank's base is 20 cm by 16 cm.
  • The water starts 25 cm deep and ends 30 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
30−25=530 - 25 = 5
The water climbed 5 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 20 by 16 base, 5 cm thick.
20×16×5=160020 \times 16 \times 5 = 1600
The stone is 1600 cm3.
Answer: 1600 cm³
4 · Reviewdoes it hold up?

The water before was 8000 cm3 and after is 9600 cm3; the difference, 1600 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 9600 minus 8000, the same 1600 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 8 medium answer: 1584 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 30 cm30\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 26 cm 22 cm 18 cm
Show solution
1 · Understandwhat's really being asked

A tank 22 cm by 18 cm holds water 26 cm deep. A stone goes in, fully under, and the water reaches 30 cm. We want the stone's volume.

Givens
  • The tank's base is 22 cm by 18 cm.
  • The water starts 26 cm deep and ends 30 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
30−26=430 - 26 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 22 by 18 base, 4 cm thick.
22×18×4=158422 \times 18 \times 4 = 1584
The stone is 1584 cm3.
Answer: 1584 cm³
4 · Reviewdoes it hold up?

The water before was 10296 cm3 and after is 11880 cm3; the difference, 1584 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 11880 minus 10296, the same 1584 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 9 hard answer: 3080 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 28 cm28\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 24 cm 35 cm 22 cm
Show solution
1 · Understandwhat's really being asked

A tank 35 cm by 22 cm holds water 24 cm deep. A stone goes in, fully under, and the water reaches 28 cm. We want the stone's volume.

Givens
  • The tank's base is 35 cm by 22 cm.
  • The water starts 24 cm deep and ends 28 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
28−24=428 - 24 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 35 by 22 base, 4 cm thick.
35×22×4=308035 \times 22 \times 4 = 3080
The stone is 3080 cm3.
Answer: 3080 cm³
4 · Reviewdoes it hold up?

The water before was 18480 cm3 and after is 21560 cm3; the difference, 3080 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 21560 minus 18480, the same 3080 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 10 hard answer: 3000 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 33 cm33\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 30 cm 40 cm 25 cm
Show solution
1 · Understandwhat's really being asked

A tank 40 cm by 25 cm holds water 30 cm deep. A stone goes in, fully under, and the water reaches 33 cm. We want the stone's volume.

Givens
  • The tank's base is 40 cm by 25 cm.
  • The water starts 30 cm deep and ends 33 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
33−30=333 - 30 = 3
The water climbed 3 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 40 by 25 base, 3 cm thick.
40×25×3=300040 \times 25 \times 3 = 3000
The stone is 3000 cm3.
Answer: 3000 cm³
4 · Reviewdoes it hold up?

The water before was 30000 cm3 and after is 33000 cm3; the difference, 3000 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 33000 minus 30000, the same 3000 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 11 hard answer: 3780 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 38 cm38\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 35 cm 45 cm 28 cm
Show solution
1 · Understandwhat's really being asked

A tank 45 cm by 28 cm holds water 35 cm deep. A stone goes in, fully under, and the water reaches 38 cm. We want the stone's volume.

Givens
  • The tank's base is 45 cm by 28 cm.
  • The water starts 35 cm deep and ends 38 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
38−35=338 - 35 = 3
The water climbed 3 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 45 by 28 base, 3 cm thick.
45×28×3=378045 \times 28 \times 3 = 3780
The stone is 3780 cm3.
Answer: 3780 cm³
4 · Reviewdoes it hold up?

The water before was 44100 cm3 and after is 47880 cm3; the difference, 3780 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 47880 minus 44100, the same 3780 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.
Variant 12 hard answer: 6000 cm³

A tank shaped like a rectangular prism holds water, as shown at the right. When a stone is placed in it so that the stone is completely under water, the water rises to a height of 44 cm44\ \text{cm}. What is the volume of the stone, in cm3\text{cm}^3? (Ignore the thickness of the tank.)

stone 40 cm 50 cm 30 cm
Show solution
1 · Understandwhat's really being asked

A tank 50 cm by 30 cm holds water 40 cm deep. A stone goes in, fully under, and the water reaches 44 cm. We want the stone's volume.

Givens
  • The tank's base is 50 cm by 30 cm.
  • The water starts 40 cm deep and ends 44 cm deep.
  • The stone is completely under water.
Unknowns
  • The volume of the stone.
Constraints
  • No water is added or spilled, and the stone's shape is never given.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The stone cannot be measured -- no shape is given. It does not need to be: the water it pushed out of the way has the same volume, and that water is a plain slab with the tank's base and the rise as its height.

3 · Execute3 carry out the plan

1Find how far the water rose

#7 Identify Subproblems 6.G.A.2
The difference between the two levels.
44−40=444 - 40 = 4
The water climbed 4 cm.

2See what that rise is made of

#16 Count the Complement 6.G.A.2
The stone took up room at the bottom, so the same amount of water had to go somewhere: straight up.
stone=water pushed up\text{stone} = \text{water pushed up}
Volume in equals volume up.

3Measure that slab of water

#17 Visualize Spatial Relationships 6.G.A.2
It fills the tank's whole 50 by 30 base, 4 cm thick.
50×30×4=600050 \times 30 \times 4 = 6000
The stone is 6000 cm3.
Answer: 6000 cm³
4 · Reviewdoes it hold up?

The water before was 60000 cm3 and after is 66000 cm3; the difference, 6000 cm3, is exactly the space the stone is taking up.

Another way: Working with the two totals rather than the rise gives 66000 minus 60000, the same 6000 cm3 with larger numbers.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Measuring the displaced water as a rectangular prism.
💡Takeaway. You do not need a shape to measure a thing. If it goes under water, the water tells you how big it is.