Free Math Worksheets

- Check whether each choice describes the entire solid or only selected parts. - Pay attention to the difference between “every” and “several.” - Decide whether the condition must hold at all heights.

1. Cavalieri's principle requires equal heights and equal areas for all corresponding cross sections parallel to the base plane. 2. Only choice B states both requirements completely.

B

- Compare the oblique prism with a right prism having the same base and perpendicular height. - Think about corresponding cross sections parallel to the bases. - Distinguish the perpendicular height from a slanted lateral edge.

1. By Cavalieri's principle, a right prism and an oblique prism with equal base areas and equal perpendicular heights have equal volumes. 2. Therefore, \(V=Bh=36\cdot14=504\,\text{cm}^3\). 3. The \(17\,\text{cm}\) lateral-edge length is not needed because volume uses the perpendicular height.

The volume is \(504\,\text{cm}^3\). The \(17\,\text{cm}\) lateral-edge length is not needed.

- Compare the oblique cylinder with a right cylinder having the same base and perpendicular height. - Convert the diameter to a radius. - Convert cubic centimeters to cubic decimeters at the end.

1. By Cavalieri's principle, the oblique cylinder has the same volume as a right cylinder with the same circular base and perpendicular height. 2. The radius is \(6\,\text{cm}\), so \(V=\pi r^2h=\pi\cdot6^2\cdot15=540\pi\,\text{cm}^3\). 3. Since \(1{,}000\,\text{cm}^3=1\,\text{dm}^3\), \(V=0.54\pi\,\text{dm}^3\approx1.70\,\text{dm}^3\). 4. The \(18\,\text{cm}\) slanted edge does not affect the volume.

The volume is \(0.54\pi\,\text{dm}^3\approx1.70\,\text{dm}^3\).

- Compare the oblique cone with a right cone having the same base and perpendicular height. - The perpendicular distance from the base plane to the apex is the relevant height. - Decide whether shifting the apex sideways changes corresponding parallel cross sections.

1. An oblique cone and a right cone with the same base and perpendicular height have equal cross-sectional areas at corresponding heights. 2. By Cavalieri's principle, their volumes are equal. 3. Therefore, \(V=\frac{1}{3}\pi r^2h=\frac{1}{3}\pi\cdot4.5^2\cdot12=81\pi\,\text{cm}^3\approx254.47\,\text{cm}^3\).

The volume is \(81\pi\,\text{cm}^3\approx254.47\,\text{cm}^3\).

- Identify what changes and what remains unchanged when the cards are shifted. - Compare horizontal cross sections of the shifted and unshifted stacks. - Use the volume of the corresponding right prism.

1. Every horizontal cross section remains a square with area \(8\cdot8=64\,\text{cm}^2\). 2. By Cavalieri's principle, the shifted stack has the same volume as an unshifted rectangular prism with the same cross-sectional area and height. 3. Thus, \(V=64\cdot22=1{,}408\,\text{cm}^3\).

The volume is \(1{,}408\,\text{cm}^3\).

- Compare the column with a simpler solid having the same cross-sectional areas. - The area of each cross section matters more than its exact shape. - Use the common height.

1. Compare the column with a right prism that is \(10\,\text{cm}\) tall and has constant cross-sectional area \(18\,\text{cm}^2\). 2. The two solids have equal cross-sectional areas at every height, so Cavalieri's principle gives equal volumes. 3. Thus, \(V=18\cdot10=180\,\text{cm}^3\).

The column volume is \(180\,\text{cm}^3\).

- Compare all corresponding cross sections, not just one. - A constant cross-sectional area factor produces the same volume factor when heights are equal. - Apply the factor to the known volume.

1. Because the same area scale factor applies to all corresponding cross sections and the heights are equal, the volume has the same scale factor. 2. Therefore, \(V_B=1.25V_A=1.25\cdot320=400\,\text{cm}^3\).

\(V_B=400\,\text{cm}^3\)

- Transfer the ratio of corresponding cross-sectional areas to the volumes. - Identify which part of the ratio corresponds to the known volume. - The unknown volume should be larger than \(270\,\text{cm}^3\).

1. By Cavalieri's principle, a constant ratio of \(3\) to \(2\) for corresponding cross-sectional areas gives the same ratio of volumes. 2. The given \(270\,\text{cm}^3\) corresponds to the smaller, \(2\)-part volume. 3. Therefore, \(V=270\left(\frac{3}{2}\right)=405\,\text{cm}^3\).

The other solid has volume \(405\,\text{cm}^3\).

- Determine the cylinder's cross-sectional area. - Because the heights are equal, compare the cross-sectional areas directly. - The unknown is an area, not a volume.

1. Both solids have constant cross-sectional area throughout their height. 2. To make corresponding cross sections equal, the prism base area must equal the cylinder's circular base area. 3. Therefore, \(B=\pi r^2=\pi\cdot3^2=9\pi\,\text{cm}^2\approx28.27\,\text{cm}^2\).

\(B=9\pi\,\text{cm}^2\approx28.27\,\text{cm}^2\)

- Find the known prism volume first. - Replace the unusual column mentally with a prism having the same cross-sectional area. - Work backward from volume to height.

1. The prism volume is \(V=50\cdot12=600\,\text{cm}^3\). 2. By Cavalieri's principle, the column has the same volume as a prism with base area \(40\,\text{cm}^2\) and the same height as the column. 3. Let the column height be \(h\). Then \(40h=600\), so \(h=15\,\text{cm}\).

The column is \(15\,\text{cm}\) tall.

- Decide whether information about only the bottom of a solid determines the entire solid. - Think of solids that narrow at different rates. - State a condition that applies throughout the full height.

1. The claim is false for arbitrary solids because equal base areas and equal heights do not determine the areas of the other cross sections. 2. The missing condition is that, at every corresponding height, cross sections parallel to the base planes must have equal areas.

The claim is false. The solids must have equal heights and equal areas for corresponding cross sections at every height.

- Compare the two area expressions at an arbitrary height. - Distinguish cross-sectional shape from cross-sectional area. - Confirm that both solids use the same height interval.

1. For every \(z\) from \(0\) to \(8\), the two cross-sectional areas are equal. 2. The solids also have equal heights. 3. By Cavalieri's principle, \(V_K=V_L\). The different cross-sectional shapes do not matter because their areas are equal.

Yes. The volumes are equal: \(V_K=V_L\).

- For a cross section made of separate regions, add their areas. - Compare the total cross-sectional areas of the two solids. - Use a simple comparison solid with the same height and constant cross-sectional area.

1. At every height, the total cross-sectional area of solid \(B\) is \(12\,\text{cm}^2+23\,\text{cm}^2=35\,\text{cm}^2\). 2. Thus, solids \(A\) and \(B\) have equal cross-sectional areas at every corresponding height and have the same height. 3. By Cavalieri's principle, their volumes are equal. Using a prism with base area \(35\,\text{cm}^2\) and height \(9\,\text{cm}\), \(V_A=V_B=35\cdot9=315\,\text{cm}^3\).

Both solids have volume \(315\,\text{cm}^3\).

- Determine how much of each cross section is actually material. - Subtract the shaft area from the outer area. - Compare the result with a prism having a constant cross-sectional area.

1. The material area in each horizontal cross section is \(64\,\text{m}^2-16\,\text{m}^2=48\,\text{m}^2\). 2. A solid prism with base area \(48\,\text{m}^2\) and height \(20\,\text{m}\) has the same cross-sectional area at every corresponding height. 3. By Cavalieri's principle, the material volume is \(V=48\cdot20=960\,\text{m}^3\).

The volume of material is \(960\,\text{m}^3\).

- Replace the offset duct with a right comparison prism. - Use the fact that corresponding cross sections have equal areas. - Convert the volume only after calculating it.

1. The offset duct has the same volume as a right prism with base area \(0.08\,\text{m}^2\) and height \(2.5\,\text{m}\). 2. The volume is \(V=0.08\cdot2.5=0.2\,\text{m}^3\). 3. Since \(1\,\text{m}^3=1{,}000\,\text{dm}^3\), \(0.2\,\text{m}^3=200\,\text{dm}^3\).

The interior volume is \(200\,\text{dm}^3\).

- Compare the container with a right prism having the same parallel cross sections. - Use the perpendicular distance, not the length of an oblique edge. - Check the unit used for capacity.

1. The oblique container has the same volume as a right prism with the same end area and the same perpendicular height. 2. Therefore, \(V=1.8\cdot6=10.8\,\text{m}^3\).

The capacity is \(10.8\,\text{m}^3\).

- Use the stated standard solid as a comparison. - The sculpture's outer shape does not matter when corresponding cross-sectional areas are equal. - Calculate the comparison solid's volume.

1. The sculpture and the cone have the same height and equal cross-sectional areas at every corresponding height, so Cavalieri's principle gives them equal volumes. 2. The cone's volume is \(V=\frac{1}{3}\pi\cdot6^2\cdot15=180\pi\,\text{cm}^3\approx565.49\,\text{cm}^3\).

The volume is \(180\pi\,\text{cm}^3\approx565.49\,\text{cm}^3\).

- Decide whether the redesign changes the area of each layer. - Compare the original and new layers at the same height. - Calculate the percent change only after comparing the volumes.

1. The original and redesigned components have equal heights and equal corresponding cross-sectional areas, so Cavalieri's principle gives equal volumes. 2. The new volume is \(2.4\,\text{m}^3\). 3. The volume change is \(2.4-2.4=0\,\text{m}^3\), so the percent change is \(0\%\).

The new volume is \(2.4\,\text{m}^3\), and the percent change is \(0\%\).

- Read the dimensions of both cross sections from the diagram. - Compare their areas, not just their shapes. - Then use the common height of the solids.

1. For solid \(K\), each cross section is a rectangle with area \(A_K=8\cdot3=24\,\text{cm}^2\). 2. For solid \(L\), each cross section is a triangle with area \(A_L=\frac{1}{2}\cdot12\cdot4=24\,\text{cm}^2\). 3. The solids have equal heights and equal cross-sectional areas at every corresponding height, so Cavalieri's principle gives equal volumes. 4. Therefore, \(V_K=V_L=24\cdot15=360\,\text{cm}^3\).

The volumes are equal: \(V_K=V_L=360\,\text{cm}^3\).

- In b), pay attention to the dashed segment. - Compare the base and corresponding height of the two triangles. - Then use the fact that these cross sections occur at every height of the solids.

1. The triangles in a) and b) have the same base, \(10\,\text{cm}\), and the same corresponding height, \(6\,\text{cm}\). 2. Therefore, every cross section in each solid has area \(A=\frac{1}{2}\cdot10\cdot6=30\,\text{cm}^2\). 3. By Cavalieri's principle, \(V_a=V_b=30\cdot16=480\,\text{cm}^3\).

Both volumes are \(480\,\text{cm}^3\).

- Replace the column with a simple solid having the same cross sections. - Work backward from volume to area.

1. The column has the same volume as a prism with unknown base area \(G\) and height \(18\,\text{cm}\). 2. From \(18G=936\), \(G=52\,\text{cm}^2\).

Each cross section has area \(52\,\text{cm}^2\).

- Compare cross sections taken the same distance from each apex. - Consider how cross-sectional area scales in a pyramid. - Once volume equality is established, calculate the volume only once.

1. At the same fractional distance from the apex, corresponding cross sections of all three pyramids have the same area scale factor. 2. Because the pyramids have equal base areas and equal heights, corresponding cross sections have equal areas. By Cavalieri's principle, the volumes are equal. 3. The common volume is \(V=\frac{1}{3}Bh=\frac{1}{3}\cdot48\cdot9=144\,\text{cm}^3\).

All three pyramids have volume \(144\,\text{cm}^3\).

- Compare cross sections near the base and near the top for each pair. - Determine whether cross-sectional area changes with height. - Equal base areas alone are not sufficient.

1. For a), all corresponding cross sections have equal areas, so the volumes are equal. 2. For b), prism cross sections remain constant while pyramid cross sections decrease toward the apex, so Cavalieri's principle does not establish equal volume. 3. For c), corresponding cross sections of the right and oblique cones have equal areas, so the volumes are equal. 4. For d), cylinder and cone cross sections do not have equal areas at every height, so the principle does not establish equal volume.

a) yes b) no c) yes d) no

- Simplify and compare the area expressions pairwise. - Look for a cross-sectional area difference that remains constant. - Interpret a constant area difference as the cross section of an added prism.

1. For every height, \(A_K(z)=A_L(z)\), so Cavalieri's principle gives \(V_K=V_L\). 2. Also, \(A_M(z)-A_K(z)=1\,\text{cm}^2\) and \(A_M(z)-A_L(z)=1\,\text{cm}^2\) at every height. 3. This constant cross-sectional area difference over \(10\,\text{cm}\) corresponds to a volume difference of \(1\cdot10=10\,\text{cm}^3\). 4. Therefore, \(V_M\) is \(10\,\text{cm}^3\) greater than each of \(V_K\) and \(V_L\).

\(V_K=V_L\). The volume \(V_M\) is \(10\,\text{cm}^3\) greater than each of them.

- Find the area of one cross section of the pipe's material. - Subtract the inner circular area from the outer circular area. - Compare the oblique pipe with a right hollow cylinder of the same perpendicular height.

1. Every cross section parallel to the ends is an annulus with area \(A=\pi\left(5^2-3^2\right)=16\pi\,\text{cm}^2\). 2. The oblique pipe has the same volume as a right hollow cylinder with the same perpendicular height and annular cross sections. 3. Therefore, \(V=(16\pi)(18)=288\pi\,\text{cm}^3\approx904.78\,\text{cm}^3\).

The volume of material is \(288\pi\,\text{cm}^3\approx904.78\,\text{cm}^3\).

- Identify a familiar solid with the same cross-sectional area at every height. - The changing cross-sectional shape does not matter if the area remains equal. - Calculate in one unit first, then convert at the end.

1. The sculpture has the same volume as a cylinder with radius \(6\,\text{cm}\) and height \(20\,\text{cm}\). 2. Its volume is \(V=\pi\cdot6^2\cdot20=720\pi\,\text{cm}^3\). 3. Since \(1{,}000\,\text{cm}^3=1\,\text{dm}^3\), \(720\pi\,\text{cm}^3=0.72\pi\,\text{dm}^3\approx2.26\,\text{dm}^3\).

The volume is \(0.72\pi\,\text{dm}^3\approx2.26\,\text{dm}^3\).

- Compare cross sections at the same perpendicular distance from the base. - Decide whether shifting the apex changes the area scale factor of those cross sections. - Calculate the common volume only once.

1. At the same perpendicular distance from the base, a cross section parallel to the base has the same area in both pyramids. Shifting the apex sideways changes the position of the cross section, but not its area scale factor. 2. The pyramids have equal heights and equal corresponding cross-sectional areas, so Cavalieri's principle shows that their volumes are equal. 3. Their common volume is \(V=\frac{1}{3}\cdot80\cdot9=240\,\text{cm}^3\).

Both pyramids have volume \(240\,\text{cm}^3\).

- Compare cross sections near the base and near the top. - Determine whether the cross-sectional area stays constant in both solids. - Equal base area and equal height alone are not sufficient.

1. The corresponding cross-sectional areas are not equal at every height. A prism's cross-sectional area remains constant, while a pyramid's cross-sectional area decreases toward the apex. 2. Therefore, equal base area and equal height do not satisfy the cross-section condition of Cavalieri's principle for these two solids. 3. The prism's volume is \(V_{\text{prism}}=54\cdot10=540\,\text{cm}^3\). 4. The pyramid's volume is \(V_{\text{pyramid}}=\frac{1}{3}\cdot54\cdot10=180\,\text{cm}^3\).

Nora's reasoning is incorrect. The prism has volume \(540\,\text{cm}^3\), and the pyramid has volume \(180\,\text{cm}^3\).

- Treat the difference between the two cross sections as a separate simple solid. - This area difference remains constant over the entire height. - Add the resulting extra volume to the known volume.

1. The constant cross-sectional area difference can be represented by an added prism with base area \(7\,\text{cm}^2\) and height \(12\,\text{cm}\). 2. The added volume is \(7\cdot12=84\,\text{cm}^3\). 3. Therefore, \(V_A=420+84=504\,\text{cm}^3\).

The volume of solid \(A\) is \(504\,\text{cm}^3\).

- Compare the constant cross-section scale factor with the volume scale factor. - Write an equation for the unknown value. - Check that the resulting factor is consistent with the larger volume.

1. Because the corresponding cross-sectional areas differ by a constant factor, the volumes differ by the same factor. 2. Therefore, \(\frac{x}{5}=1.4\). 3. Multiplying by \(5\) gives \(x=7\).

\(x=7\).

- Use all the dimensions shown in the diagram. - Compare the areas of the two different shapes. - Determine what equal cross-sectional areas at every height imply.

1. The trapezoidal cross section in a) has area \(A_a=\frac{10+6}{2}\cdot4=32\,\text{cm}^2\). 2. The rectangular cross section in b) has area \(A_b=8\cdot4=32\,\text{cm}^2\). 3. The solids have equal heights and equal cross-sectional areas at every height, so Cavalieri's principle gives \(V_a=V_b=32\cdot9=288\,\text{cm}^3\).

Both volumes are \(288\,\text{cm}^3\).

- Distinguish the material from the hollow region in the diagram. - Find the total material area of each cross section. - Then use the common height.

1. In a), the material area is \(12\cdot8-8\cdot4=96-32=64\,\text{cm}^2\). 2. In b), the material area is \(8\cdot8=64\,\text{cm}^2\). 3. The corresponding material cross sections have equal areas at every height, so both components contain \(64\cdot20=1{,}280\,\text{cm}^3\) of material.

Both components contain \(1{,}280\,\text{cm}^3\) of material.

- Find the three cross-sectional areas one at a time. - Because the solids have equal heights, compare their cross-sectional areas directly. - Check which solid has the largest cross section.

1. The cross-sectional areas are \(A_a=5\cdot5=25\,\text{cm}^2\), \(A_b=10\cdot2.5=25\,\text{cm}^2\), and \(A_c=\frac{1}{2}\cdot12\cdot5=30\,\text{cm}^2\). 2. Since a) and b) have equal cross-sectional areas at every height and the same height, Cavalieri's principle gives equal volumes. 3. The volumes are \(V_a=V_b=25\cdot12=300\,\text{cm}^3\) and \(V_c=30\cdot12=360\,\text{cm}^3\).

Solids a) and b) have equal volumes of \(300\,\text{cm}^3\) each. Solid c) has volume \(360\,\text{cm}^3\).

- Distinguish the outer region from the hollow space. - Compare the resulting areas symbolically. - Keep the factor \(\pi\) until the final approximation.

1. The annular material area in a) is \(\pi\left(5^2-3^2\right)=16\pi\,\text{cm}^2\). 2. The circular material area in b) is \(\pi\cdot4^2=16\pi\,\text{cm}^2\). 3. The material cross sections have equal areas at every height, so Cavalieri's principle gives \(V=(16\pi)(11)=176\pi\,\text{cm}^3\approx552.92\,\text{cm}^3\).

Both material volumes are \(176\pi\,\text{cm}^3\approx552.92\,\text{cm}^3\).

- In b), add the areas of the two separate regions. - Compare that total with the area in a). - Separating the regions does not change their total area.

1. In a), the triangular area is \(A_A=\frac{1}{2}\cdot12\cdot8=48\,\text{cm}^2\). 2. In b), each triangle has area \(\frac{1}{2}\cdot6\cdot8=24\,\text{cm}^2\), so the total cross-sectional area is \(48\,\text{cm}^2\). 3. The solids have equal total cross-sectional areas at every height and equal heights. By Cavalieri's principle, \(V_A=V_B=48\cdot10=480\,\text{cm}^3\).

Both solids have volume \(480\,\text{cm}^3\).

- First find the fully dimensioned cross-sectional area. - Use the equal volumes together with the common height. - Then calculate the common volume only once.

1. Because the solids have equal heights, equal volumes, and constant cross-sectional areas, those constant areas must be equal. 2. The triangular cross section in b) has area \(\frac{1}{2}\cdot12\cdot5=30\,\text{cm}^2\). 3. For a), \(6x=30\), so \(x=5\,\text{cm}\). 4. The common volume is \(V=30\cdot13=390\,\text{cm}^3\).

\(x=5\,\text{cm}\), and the common volume is \(390\,\text{cm}^3\).

- Calculate both cross-sectional areas. - Simplify the resulting ratio completely. - For equal heights, the cross-sectional area ratio equals the volume ratio.

1. From the diagram, \(A_A=9\cdot4=36\,\text{cm}^2\) and \(A_B=\frac{1}{2}\cdot12\cdot9=54\,\text{cm}^2\). 2. Thus, \(\frac{V_A}{V_B}=\frac{A_A}{A_B}=\frac{36}{54}=\frac{2}{3}\) because the solids have the same height. 3. The volumes are \(V_A=36\cdot8=288\,\text{cm}^3\) and \(V_B=54\cdot8=432\,\text{cm}^3\).

The ratio of \(V_A\) to \(V_B\) is \(2\) to \(3\). Also, \(V_A=288\,\text{cm}^3\) and \(V_B=432\,\text{cm}^3\).

- Test the claim by calculating all three cross-sectional areas. - For the parallelogram, use the dashed height. - Compare the volumes only after comparing the areas.

1. The cross-sectional areas are \(A_a=8\cdot5=40\,\text{cm}^2\), \(A_b=10\cdot4=40\,\text{cm}^2\), and \(A_c=\frac{1}{2}\cdot10\cdot8=40\,\text{cm}^2\). 2. Ben's claim is false. The triangle's greater height offsets the factor \(\frac{1}{2}\) in its area formula. 3. All three solids have equal cross-sectional areas at every height and equal heights, so Cavalieri's principle gives \(V=40\cdot7=280\,\text{cm}^3\) for each solid.

Ben's claim is false. All three volumes are \(280\,\text{cm}^3\).

- Calculate the reference cross-sectional area first. - Test each candidate separately. - Because the heights are equal, the cross-sectional areas determine the comparison.

1. The reference cross-sectional area is \(A_R=7\cdot4=28\,\text{cm}^2\). 2. Candidate a) has area \(\frac{1}{2}\cdot14\cdot4=28\,\text{cm}^2\), candidate b) has area \(\frac{9+5}{2}\cdot4=28\,\text{cm}^2\), and candidate c) has area \(5\cdot5=25\,\text{cm}^2\). 3. Therefore, a) and b) have the same volume as \(R\): \(V_R=V_a=V_b=28\cdot9=252\,\text{cm}^3\). 4. Candidate c) has volume \(V_c=25\cdot9=225\,\text{cm}^3\).

Candidates a) and b) have the same volume as \(R\). \(V_R=V_a=V_b=252\,\text{cm}^3\), and \(V_c=225\,\text{cm}^3\).

- Use the diagram to find both cross-sectional areas. - Calculate the known volume first. - Then work backward to find the unknown height.

1. The cross-sectional areas are \(A_A=9\cdot4=36\,\text{cm}^2\) and \(A_B=\frac{1}{2}\cdot10\cdot6=30\,\text{cm}^2\). 2. By Cavalieri's principle, each solid has the same volume as a right prism with the same constant cross-sectional area and height. Thus, \(V_A=36\cdot10=360\,\text{cm}^3\) and \(V_B=30h\). 3. For equal volumes, \(30h=360\). Therefore, \(h=12\,\text{cm}\).

Solid \(B\) must be \(12\,\text{cm}\) tall.

- Consider how a height changes when a solid is turned upside down. - Convert complementary heights into corresponding heights. - Changing a solid's orientation does not change its volume.

1. Turn solid \(L\) upside down. A cross section that was originally at height \(12-z\) is then at height \(z\). 2. Therefore, at every corresponding height \(z\), solid \(K\) and the inverted solid \(L\) have equal cross-sectional areas. 3. Turning a solid over does not change its volume, so Cavalieri's principle gives \(V_K=V_L\).

After solid \(L\) is turned upside down, all corresponding cross-sectional areas are equal. Therefore, \(V_K=V_L\).

- Identify which length measures the distance between the base planes. - Compare the oblique cylinder with a right cylinder having the same perpendicular height. - Find the error amount only after calculating the correct volume.

1. Tim used the slant edge length instead of the perpendicular height between the bases. The corresponding right cylinder has height \(12\,\text{cm}\). 2. The correct volume is \(V=\pi\cdot5^2\cdot12=300\pi\,\text{cm}^3\approx942.48\,\text{cm}^3\). 3. Tim's result is \(325\pi\,\text{cm}^3\). The difference is \(325\pi-300\pi=25\pi\,\text{cm}^3\approx78.54\,\text{cm}^3\).

The correct volume is \(300\pi\,\text{cm}^3\approx942.48\,\text{cm}^3\). Tim's result is too large by \(25\pi\,\text{cm}^3\approx78.54\,\text{cm}^3\).

- Add the two cross-sectional areas at the same height. - Check whether the height-dependent terms cancel. - Interpret the resulting total cross-sectional area using a comparison solid.

1. At each height, the sum of the cross-sectional areas is \(A_A(z)+A_B(z)=10+2z+28-2z=38\,\text{cm}^2\). 2. Together, the two solids have the same total cross-sectional area at every height as a prism with base area \(38\,\text{cm}^2\) and height \(9\,\text{cm}\). 3. Therefore, \(V_A+V_B=38\cdot9=342\,\text{cm}^3\).

\(V_A+V_B=342\,\text{cm}^3\).

- Calculate the two volumes independently. - Then compare at least one pair of corresponding cross sections. - A true implication does not necessarily remain true when reversed.

1. The prism's volume is \(V_{\text{prism}}=30\cdot10=300\,\text{cm}^3\). 2. The pyramid's volume is \(V_{\text{pyramid}}=\frac{1}{3}\cdot90\cdot10=300\,\text{cm}^3\). 3. However, their base cross-sectional areas are already different: \(30\,\text{cm}^2\) for the prism and \(90\,\text{cm}^2\) for the pyramid. 4. Thus, equal height and equal volume do not imply equal cross-sectional areas at every corresponding height. The converse is false.

The claim is false. Both volumes are \(300\,\text{cm}^3\), but the corresponding cross-sectional areas are not equal at every height.