First, find the volume of the original 3-layer box: $$4 \times 3 \times 3 = 36$$ unit cubes. Then add the fourth layer: $$4 \times 3 = 12$$ more cubes, giving $$36 + 12 = 48$$ cubes. Finally, subtract the 2 removed cubes: $$48 - 2 = 46$$ unit cubes. Choice A subtracts 2 from the original volume incorrectly. Choice B represents only the first 3 layers. Choice D ignores the removal of 2 cubes.

When you encounter volume problems with two different methods giving different answers, you need to evaluate which method is more reliable and why discrepancies might occur.

Let's check Method 2 first: $$7 \times 3 \times 4 = 84$$ cubic units. This calculation is mathematically correct. The formula length × width × height always gives the accurate volume for rectangular prisms because it counts every unit cube systematically - you're finding how many cubes fit in each layer (7 × 3 = 21) and multiplying by the number of layers (4).

Method 1 involves manually counting individual cubes, which is prone to human error, especially with larger numbers. When counting 84 separate cubes, it's easy to lose track, double-count, or miss a few cubes.

Looking at the wrong answers: Choice A suggests Method 2 doesn't work for cube arrangements, but the length × width × height formula is universally valid for rectangular prisms. Choice B claims the dimensions were wrong, but if that were true, Method 2 would still be internally consistent with whatever dimensions were used. Choice C states Method 1 is more accurate, but systematic calculation is always more reliable than manual counting for this type of problem.

Choice D correctly identifies that the student made a counting error - missing 2 cubes out of 84 is a very reasonable mistake when counting manually.

Study tip: When two methods for finding volume disagree, trust the mathematical calculation over manual counting. The formula eliminates human error and is the standard method taught because of its reliability.

The base layer has an area of 18 square units with length 6 units. Using Area = length × width, we get $$18 = 6 \times w$$, so $$w = 3$$ units. We can verify: the total volume is $$6 \times 3 \times 4 = 72$$ cubic units. Choice A results from dividing by 9 instead of 6. Choice D uses the length as the width. Choice C confuses the height with the width dimension.

The volume of a rectangular prism can be found by packing it with unit cubes and counting how many fit inside without gaps or overlaps. In this prism, the base has 3 rows with 8 cubes in each row, making 24 cubes per layer, and there are 2 layers stacked straight up to form the height. This means you can multiply the number of cubes in one layer by the number of layers: 24 × 2 = 48, which is the same as multiplying length × width × height. Having no gaps or overlaps ensures that every part of the prism is accounted for exactly once, giving an accurate volume measurement. A common misconception is to add the layers instead of multiplying, but multiplication accounts for the total cubes. Packing with unit cubes visually demonstrates how the volume is structured as layers of area stacked to a certain height. This method helps understand that the volume formula V = l × w × h comes from the number of unit cubes along each dimension.

When you encounter problems about rotating 3D objects, remember that volume is an intrinsic property that doesn't change based on orientation—just like how a box of cereal contains the same amount whether it's standing upright or lying on its side.

To find the volume of any rectangular container, you multiply length × width × height. With the unit cubes arranged in a $$5 \times 4 \times 3$$ pattern, the volume is $$5 \times 4 \times 3 = 60$$ cubic units. When you rotate the container so the $$4 \times 3$$ face becomes the base, you're simply changing which dimension you call "height"—but all three dimensions still exist. The container still measures 5 units, 4 units, and 3 units in its three directions, just oriented differently.

Choice A is correct because rotation never changes volume—the same 60 unit cubes are still packed inside the container.

Choice B (72 cubic units) might tempt you if you mistakenly think rotation somehow adds space, perhaps by incorrectly calculating $$4 \times 3 \times 6$$ or making an arithmetic error.

Choice C (48 cubic units) could result from incorrectly thinking some dimension disappears during rotation, possibly calculating $$4 \times 3 \times 4$$ instead of using all three original dimensions.

Choice D suggests the problem lacks information, but you have everything needed—the three dimensions of the rectangular arrangement.

Study tip: Volume problems involving rotation are testing whether you understand that volume is independent of orientation. The same object always has the same volume regardless of how it's positioned in space.

Student 1's volume: $$2 \times 3 \times 8 = 48$$ cubic units. Student 2's volume: $$4 \times 4 \times 3 = 64$$ cubic units. Student 2's prism holds more cubes despite being shorter, showing that total volume depends on all three dimensions, not just height. Choice A and B incorrectly focus only on height. Choice D miscalculates Student 2's volume as 48 instead of 64.

Box A volume: $$6 \times 4 \times 3 = 72$$ cubic units. Box B volume: $$6 \times 6 \times 2 = 72$$ cubic units. Both boxes contain the same number of unit cubes arranged differently, so they have equal volumes of 72 cubic units. This demonstrates that volume depends on total unit cubes, not arrangement. Choice A and B incorrectly focus on layers rather than total volume. Choice D introduces irrelevant factors.

When you encounter problems about rectangular prisms and volume, remember that volume equals length × width × height, and you need to use the given relationships between dimensions to find each measurement.

Let's work through this step-by-step. You're told the width is 2 units, the height is 2 units more than the width, and the length is twice the width. So:

• Width = 2 units
• Height = 2 + 2 = 4 units
• Length = 2 × 2 = 4 units

The volume of this rectangular prism is $$2 × 4 × 4 = 32$$ cubic units. Since Marcus has 36 unit cubes but the arrangement only holds 32 cubes, not all his cubes will fit.

Looking at the wrong answers: Choice A calculates the volume as 24 cubes, which might result from using incorrect dimensions like 2 × 3 × 4. Choice B claims 48 cubes fit, possibly from miscalculating as 2 × 6 × 4 or doubling one dimension incorrectly. Choice C suggests exactly 36 cubes fit, which would be convenient but doesn't match our calculation.

Choice D correctly identifies that the arrangement holds only 32 cubes, meaning Marcus will have 4 cubes left over.

Strategy tip: When solving rectangular prism problems, always write out each dimension clearly before multiplying. Double-check that you've correctly applied the relationships given in the problem (like "2 more than" or "twice the width") before calculating volume.

Using Volume = base area × height, the original height is $$60 ÷ 15 = 4$$ units. The new height is $$4 + 2 = 6$$ units. The new volume is $$15 \times 6 = 90$$ cubic units. Choice A incorrectly adds 15 to the original volume. Choice C doubles the original volume incorrectly. Choice D multiplies the base area by the added height only.

The original volume is $$5 \times 4 \times 3 = 60$$ unit cubes. Since the same cubes are rearranged, the new volume must also be 60. Using $$6 \times 2 \times h = 60$$, we get $$12h = 60$$, so $$h = 5$$ units. Choice C doubles the correct height. Choice D uses the new base area as the height. Choice B results from incorrectly dividing 60 by the new length only.