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Volume of prisms and composite solids

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5111775
Two students build different solids from unit cubes with a volume of \(1\,\text{cm}^3\) each. Lisa builds a three-level stair-step solid. The bottom level contains \(6\) cubes, the middle level contains \(4\) cubes, and the top level contains \(2\) cubes. Max builds a rectangular prism with dimensions \(3\,\text{cm}\), \(2\,\text{cm}\), and \(2\,\text{cm}\). Compare the volumes of the two solids. Does one have a greater volume, or are the volumes equal?

Hints

- How many unit cubes did Lisa use altogether? - Use the dimensions to find how many unit cubes fit in Max's rectangular prism. - Compare the two totals.

Solution

1. Lisa's solid contains \(6+4+2=12\) unit cubes, so its volume is \(12\,\text{cm}^3\). 2. Max's rectangular prism has a volume of \(3\times2\times2=12\,\text{cm}^3\). 3. The two solids have equal volumes.

Answer

The volumes are equal. Each solid has a volume of \(12\,\text{cm}^3\).
5327985
The building plan shows stacks of blue wooden cubes. Each number tells how many cubes are stacked at that location. Each cube has edge length \(5\,\text{cm}\). 1. How many cubes were used? 2. What is the volume of one cube? 3. Find the total volume of the structure.
Figure for problem 532798

Hints

- Add all the stack heights in the plan. - Use \(V=s\times s\times s\) for one cube. - Multiply the volume of one cube by the total number of cubes.

Solution

1. Add the stack heights: \(2+1+2+2+1+2=10\). The structure contains \(10\) cubes. 2. One cube has volume \(5\times5\times5=125\,\text{cm}^3\). 3. The total volume is \(10\times125=1250\,\text{cm}^3\).

Answer

1. \(10\) cubes 2. \(125\,\text{cm}^3\) 3. \(1250\,\text{cm}^3\)
5355965
A building block is made from two rectangular prisms joined together. The lower prism measures \(4\,\text{cm}\times2\,\text{cm}\times1\,\text{cm}\). A second prism measuring \(2\,\text{cm}\times2\,\text{cm}\times1\,\text{cm}\) is centered on top. Find the volume of the composite block.
Figure for problem 535596

Hints

- Find the volume of each rectangular prism. - Add the two volumes.

Solution

1. The lower prism has volume \(4\times2\times1=8\,\text{cm}^3\). 2. The upper prism has volume \(2\times2\times1=4\,\text{cm}^3\). 3. The total volume is \(8+4=12\,\text{cm}^3\).

Answer

The composite block has volume \(12\,\text{cm}^3\).
5356245
A three-level awards platform is made from three concrete rectangular prisms placed side by side. Each prism is \(30\,\text{cm}\) deep and \(10\,\text{cm}\) wide. Their heights are \(10\,\text{cm}\), \(20\,\text{cm}\), and \(30\,\text{cm}\). Find the platform's total volume.
Figure for problem 535624

Hints

- Treat the platform as three rectangular prisms. - Find the volume of each prism. - Add the three volumes.

Solution

1. The first prism has volume \(10\times30\times10=3000\,\text{cm}^3\). 2. The second prism has volume \(10\times30\times20=6000\,\text{cm}^3\). 3. The third prism has volume \(10\times30\times30=9000\,\text{cm}^3\). 4. The total volume is \(3000+6000+9000=18{,}000\,\text{cm}^3\).

Answer

The total volume is \(18{,}000\,\text{cm}^3\).
5358025
Find the volume of the T-shaped beam shown. The horizontal beam is \(60\,\text{cm}\) long, \(20\,\text{cm}\) high, and \(20\,\text{cm}\) deep. The centered vertical support is \(20\,\text{cm}\) wide, \(30\,\text{cm}\) high, and \(20\,\text{cm}\) deep.
Figure for problem 535802

Hints

- Treat the solid as two rectangular prisms. - Both parts have the same depth. - Add the two volumes.

Solution

1. The horizontal beam has volume \(60\times20\times20=24{,}000\,\text{cm}^3\). 2. The vertical support has volume \(20\times30\times20=12{,}000\,\text{cm}^3\). 3. The total volume is \(24{,}000+12{,}000=36{,}000\,\text{cm}^3\).

Answer

The volume is \(36{,}000\,\text{cm}^3\).
5318925
The diagram shows a structure made from unit cubes. a) How many unit cubes are in the structure? b) The structure will be completed to form a \(3 \times 3 \times 3\) rectangular prism. How many unit cubes must be added?
Figure for problem 531892

Hints

- Count each stack in the diagram, including cubes below the top cube. - How many unit cubes fill a \(3 \times 3 \times 3\) rectangular prism? - Subtract the number already present from the completed prism's total.

Solution

1. Count the cubes by rows. The front row contains \(1 + 1 + 0 = 2\) cubes, the middle row contains \(2 + 2 + 1 = 5\) cubes, and the back row contains \(3 + 3 + 2 = 8\) cubes. The structure contains \(2 + 5 + 8 = 15\) cubes. 2. A complete \(3 \times 3 \times 3\) rectangular prism contains \(3 \times 3 \times 3 = 27\) unit cubes. 3. The number of cubes to add is \(27 - 15 = 12\).

Answer

a) \(15\) unit cubes b) \(12\) unit cubes
5319405
The building shown is made from unit cubes with edge length \(1\,\text{cm}\). A square shaft runs through the center from the top to the bottom. What is the volume of the building in cubic centimeters?
Figure for problem 531940

Hints

- First imagine the structure as a completely filled large cube. - Find the volume of the empty shaft through the center. - Subtract the shaft’s volume from the full cube’s volume. - Each unit cube has volume \(1\,\text{cm}^3\).

Solution

1. A solid \(3 \times 3 \times 3\) cube would have volume \(3\times3\times3=27\,\text{cm}^3\). 2. The central shaft measures \(1\,\text{cm}\times1\,\text{cm}\times3\,\text{cm}\), so its volume is \(3\,\text{cm}^3\). 3. Subtract the shaft: \(27-3=24\,\text{cm}^3\).

Answer

\(24\,\text{cm}^3\)
5319545
The cube building shown is made from identical small wooden cubes. Each small cube has an edge length of \(3\,\text{cm}\). 1. How many small cubes are in the building? 2. What is the volume of one small cube? 3. What is the total volume of the building in cubic centimeters?
Figure for problem 531954

Hints

- Count the stacks carefully, including cubes hidden under other cubes. - How do you find the volume of a cube from its edge length? - Once you know the volume of one cube, how can you find the volume of the entire building?

Solution

1. Count the cubes by rows of stacks. The back row has \(3+2+1=6\) cubes, the middle row has \(2+1=3\) cubes, and the front row has \(1\) cube. The building contains \(6+3+1=10\) cubes. 2. One small cube has volume \(3\times3\times3=27\,\text{cm}^3\). 3. The total volume is \(10\times27\,\text{cm}^3=270\,\text{cm}^3\).

Answer

1. \(10\) cubes 2. \(27\,\text{cm}^3\) 3. \(270\,\text{cm}^3\)
5319585
The U-shaped structure shown is made from green wooden cubes. Each cube has edge length \(5\,\text{cm}\). a) How many cubes are in the structure? b) Find the total volume in cubic centimeters. c) Express the total volume in cubic decimeters.
Figure for problem 531958

Hints

- Count the cubes by adding the stack heights. - Find the volume of one small cube. - Use \(1\,\text{dm}^3=1000\,\text{cm}^3\) for the conversion.

Solution

1. Add the stack heights: \(1+0+1+1+0+1+2+2+2=10\). The structure contains \(10\) cubes. 2. One cube has volume \(5\times5\times5=125\,\text{cm}^3\). 3. The total volume is \(10\times125=1250\,\text{cm}^3\). 4. Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), \(1250\div1000=1.25\,\text{dm}^3\).

Answer

a) The structure contains \(10\) cubes. b) The total volume is \(1250\,\text{cm}^3\). c) The total volume is \(1.25\,\text{dm}^3\).
5328505
The unit-cube building has two steps. a) How many cubes are in the building? b) What is its volume in cubic units? c) The smallest rectangular prism that can contain the building is \(3 \times 2 \times 2\) units. How many unit cubes are missing from that prism?
Figure for problem 532850

Hints

- Count the cubes in the front and back rows separately. - Each unit cube contributes one cubic unit of volume. - Compare the building's volume with the volume of the containing rectangular prism.

Solution

1. The front row contains \(3\) cubes. The back row has three stacks of height \(2\), so it contains \(3 \times 2=6\) cubes. The building contains \(3+6=9\) cubes. 2. Because each cube has volume \(1\) cubic unit, the building's volume is \(9\) cubic units. 3. The containing rectangular prism has volume \(3 \times 2 \times 2=12\) cubic units. The number of missing cubes is \(12-9=3\).

Answer

a) \(9\) cubes b) \(9\) cubic units c) \(3\) unit cubes
5328605
A hollow square frame is built from unit cubes and is \(2\) cubes high. How many unit cubes are used altogether?
Figure for problem 532860

Hints

- Count the cubes in one layer, then double the result. - Compare a full layer with the empty square in the center.

Solution

1. A full \(4 \times 4\) layer contains \(4\times4=16\) cubes. 2. The empty \(2 \times 2\) center removes \(2\times2=4\) cubes, so one frame layer contains \(16-4=12\) cubes. 3. The frame is \(2\) cubes high, so it contains \(12\times2=24\) cubes.

Answer

The frame contains \(24\) unit cubes.
5328895
The cube building shown is made from cubes with an edge length of \(10\,\text{cm}\). Find the total volume of the building in liters.
Figure for problem 532889

Hints

- Count the cubes row by row, including cubes in each stack. - Find the volume of one cube with an edge length of \(10\,\text{cm}\). - How many cubic centimeters equal one liter?

Solution

1. The front row contains \(3\) cubes. The back row has two corner stacks with \(3\) cubes each, for \(3+3=6\) cubes. The building contains \(3+6=9\) cubes. 2. One cube has volume \(10\times10\times10=1000\,\text{cm}^3\). 3. Since \(1000\,\text{cm}^3=1\,\text{L}\), each cube has a volume of \(1\,\text{L}\). 4. The total volume is \(9\times1\,\text{L}=9\,\text{L}\).

Answer

The total volume is \(9\,\text{L}\).
5329025
A unit-cube structure with an empty center is inside a box. The black outline shows a \(3 \times 3 \times 2\) box. 1) How many unit cubes are in the structure? 2) How many cubes are needed to fill the box completely, including the center gap? 3) How many unit cubes fit in the empty box?
Figure for problem 532902

Hints

- Count the occupied positions around the empty center and determine each stack's height. - Use the three box dimensions to find its capacity. - The center gap must be filled when you find the difference.

Solution

1. The outer ring consists of \(8\) stacks with \(2\) cubes in each stack, so the structure contains \(8 \times 2 = 16\) cubes. 2. The box holds \(3 \times 3 \times 2 = 18\) unit cubes. Therefore, \(18 - 16 = 2\) cubes are missing. 3. The empty box holds \(18\) unit cubes.

Answer

1) \(16\) unit cubes 2) \(2\) unit cubes 3) \(18\) unit cubes
5329745
A larger unit-cube structure G is made from blue part B and one other part. The diagrams show part B and the complete structure G. How many unit cubes are in the other part?
Figure for problem 532974

Hints

- Count all the cubes in the complete structure G. - Count all the cubes in blue part B. - Subtract the blue-part total from the complete-structure total.

Solution

1. The complete structure G contains \(2 + 2 + 3 + 2 = 9\) unit cubes. 2. Blue part B contains \(1 + 1 + 2 + 0 = 4\) unit cubes. 3. The other part contains \(9 - 4 = 5\) unit cubes.

Answer

The other part contains \(5\) unit cubes.
5355515
A solid metal cube has edge length \(6\,\text{cm}\). A smaller cube with edge length \(2\,\text{cm}\) is cut completely from one corner, as shown. Find the volume of the remaining solid.
Figure for problem 535551

Hints

- Find the volume of the original cube. - Find the volume of the removed cube. - Subtract the removed volume from the original volume.

Solution

1. The original cube has volume \(6\times6\times6=216\,\text{cm}^3\). 2. The removed cube has volume \(2\times2\times2=8\,\text{cm}^3\). 3. The remaining volume is \(216-8=208\,\text{cm}^3\).

Answer

The remaining solid has volume \(208\,\text{cm}^3\).
5357065
Compare the volumes of solids a) and b). Do they have the same volume? Justify your answer by decomposing the solids or calculating their volumes.
Figure for problem 535706

Hints

- Split solid a) into two rectangular prisms. - Find the volume of each solid. - Compare the results.

Solution

1. Solid a) consists of a \(6\,\text{cm}\times4\,\text{cm}\times2\,\text{cm}\) prism and a \(2\,\text{cm}\times4\,\text{cm}\times4\,\text{cm}\) prism. Its volume is \(6\times4\times2+2\times4\times4=48+32=80\,\text{cm}^3\). 2. Solid b) is a \(10\,\text{cm}\times4\,\text{cm}\times2\,\text{cm}\) rectangular prism. Its volume is \(10\times4\times2=80\,\text{cm}^3\). 3. The volumes are equal.

Answer

Yes. Both solids have volume \(80\,\text{cm}^3\).
5357235
Three solids A, B, and C are shown. Solid A is a rectangular prism measuring \(10\,\text{cm}\times2\,\text{cm}\times6\,\text{cm}\). Solid B is made from prisms measuring \(8\,\text{cm}\times5\,\text{cm}\times2\,\text{cm}\) and \(4\,\text{cm}\times5\,\text{cm}\times2\,\text{cm}\). Solid C is made from prisms measuring \(12\,\text{cm}\times2\,\text{cm}\times4\,\text{cm}\) and \(4\,\text{cm}\times2\,\text{cm}\times4\,\text{cm}\). Find each volume. Which solids have equal volumes?
Figure for problem 535723

Hints

- Find the volume of solid A directly. - Split solids B and C into rectangular prisms and add their volumes. - Compare the three totals.

Solution

1. Solid A has volume \(10\times2\times6=120\,\text{cm}^3\). 2. Solid B has volume \(8\times5\times2+4\times5\times2=80+40=120\,\text{cm}^3\). 3. Solid C has volume \(12\times2\times4+4\times2\times4=96+32=128\,\text{cm}^3\). 4. Solids A and B have equal volumes.

Answer

Solid A: \(120\,\text{cm}^3\) Solid B: \(120\,\text{cm}^3\) Solid C: \(128\,\text{cm}^3\) Solids A and B have equal volumes.
5357925
An aluminum channel is \(5\,\text{cm}\) wide, \(3\,\text{cm}\) long, and \(4\,\text{cm}\) high. The bottom and both side walls are \(1\,\text{cm}\) thick. Find its volume.
Figure for problem 535792

Hints

- Divide the channel into a bottom prism and two side-wall prisms. - Use the overall height to find each wall's height above the bottom.

Solution

1. The bottom prism has volume \(5\times3\times1=15\,\text{cm}^3\). 2. Each side wall is \(4-1=3\,\text{cm}\) high and has volume \(1\times3\times3=9\,\text{cm}^3\). 3. The total volume is \(15+2\times9=33\,\text{cm}^3\).

Answer

The volume is \(33\,\text{cm}^3\).
5359095
A hollow metal block has outer dimensions \(12\,\text{cm}\times10\,\text{cm}\times6\,\text{cm}\). A rectangular opening measuring \(8\,\text{cm}\times10\,\text{cm}\times4\,\text{cm}\) passes through the block, as shown. Find the volume of metal in the block.
Figure for problem 535909

Hints

- Find the volume of the outer rectangular prism. - Subtract the volume of the opening.

Solution

1. The outer prism has volume \(12\times10\times6=720\,\text{cm}^3\). 2. The opening has volume \(8\times10\times4=320\,\text{cm}^3\). 3. The metal volume is \(720-320=400\,\text{cm}^3\).

Answer

The metal volume is \(400\,\text{cm}^3\).
5360915
A solid metal part is shaped like a rectangular prism with a notch removed from the top, as shown. Find its volume. All dimensions are in centimeters.
Figure for problem 536091

Hints

- Add the volumes of the three visible rectangular prisms. - Or subtract the notch from one larger rectangular prism.

Solution

1. One method is to add three prism volumes: \(10\times15\times20+12\times15\times12+10\times15\times20=3000+2160+3000=8160\,\text{cm}^3\). 2. Another method is to subtract the notch from the enclosing prism: \(32\times15\times20-12\times15\times8=9600-1440=8160\,\text{cm}^3\).

Answer

The volume is \(8160\,\text{cm}^3\).
5315835
The concrete staircase shown has three steps. Each step is \(2\,\text{dm}\) high and \(3\,\text{dm}\) deep. The staircase is \(10\,\text{dm}\) wide. Find its volume in cubic decimeters and in liters.
Figure for problem 531583

Hints

- Split the staircase into rectangular prisms. - Add the three partial volumes. - Relate cubic decimeters to liters.

Solution

1. Split the staircase into three horizontal rectangular-prism layers. 2. Their volumes are \(9\times10\times2=180\,\text{dm}^3\), \(6\times10\times2=120\,\text{dm}^3\), and \(3\times10\times2=60\,\text{dm}^3\). 3. The total volume is \(180+120+60=360\,\text{dm}^3\). 4. Since \(1\,\text{dm}^3=1\,\text{L}\), the staircase uses \(360\,\text{L}\) of concrete.

Answer

The volume is \(360\,\text{dm}^3\), or \(360\,\text{L}\).
5328865
A glass case can hold exactly \(4\times4\times4\) unit cubes. The blue cube building shown is inside the case. a) Find the volume of the blue building in unit cubes. b) What fraction of the case is filled with cubes? Write the fraction in simplest form. c) How many cubes must be removed so that the case is exactly half full?
Figure for problem 532886

Hints

- First determine how many unit cubes fill the entire glass case. - Count the blue building by stacks or by layers. - Compare the building's volume with the case's total volume to form a fraction. - How many cubes represent exactly half of the case?

Solution

1. The glass case can hold \(4\times4\times4=64\) unit cubes. 2. The outer frame has \(12\) stacks that are each \(2\) cubes high, for \(12\times2=24\) cubes. The inner square has \(4\) stacks that are each \(4\) cubes high, for \(4\times4=16\) cubes. The building contains \(24+16=40\) cubes. 3. The filled fraction is \(\frac{40}{64}=\frac{5}{8}\). 4. Half of the case is \(64\div2=32\) cubes. Therefore, \(40-32=8\) cubes must be removed.

Answer

a) \(40\) unit cubes b) \(\frac{5}{8}\) c) \(8\) cubes

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