Width = 3 cubes, so length = 2 × 3 = 6 cubes. Original volume = 6 × 3 × 4 = 72 cubic units. After 8 cubes fall out: 72 - 8 = 64 cubic units. Choice B is the original volume before cubes fell out. Choice C incorrectly adds 8 instead of subtracting. Choice D subtracts too many cubes (16 instead of 8).

Total cubes initially: 18 cubes per layer × 4 layers = 72 cubes. After removing 6 cracked cubes: 72 - 6 = 66 cubic centimeters. Choice A adds 6 instead of subtracting. Choice B is the original volume before removing cubes. Choice D incorrectly adds 12 to the original volume.

When you encounter problems about stacked cubes, you're working with three-dimensional thinking and volume concepts. The key is understanding that volume equals length × width × height.

Let's break down what we know: Looking from above, you see a 5 by 6 array, meaning there are 30 columns total (5 × 6 = 30). The total volume is 90 cubic units. Since each unit cube has a volume of 1 cubic unit, there are 90 individual cubes in the box.

To find how many cubes are in each column, divide the total number of cubes by the number of columns: $$90 ÷ 30 = 3$$ cubes per column.

Let's check why the other answers don't work. Choice A suggests 6 cubes per column, which would give us $$30 × 6 = 180$$ total cubes—far more than our actual 90. Choice B proposes 2 cubes per column, resulting in $$30 × 2 = 60$$ total cubes, which is 30 cubes short. Choice C suggests 4 cubes per column, giving us $$30 × 4 = 120$$ total cubes, which exceeds our target by 30 cubes.

Only choice D gives us the correct total: $$30 × 3 = 90$$ cubes.

Study tip: When solving cube-stacking problems, always identify all three dimensions. Start with what you can see (the base array), then use the total volume to find the missing dimension (height). Remember that total cubes = length × width × height when all cubes are uniform.

First, find the total volume: 4 × 3 × 2 = 24 cubic units. Then subtract the 5 cubes that were removed: 24 - 5 = 19 cubic units. Choice B is the original volume before removing cubes. Choice C adds instead of subtracts the removed cubes. Choice D represents removing 7 cubes instead of 5.

Each layer contains 4 × 3 = 12 unit cubes. Since the total volume is 36 cubic inches, the number of layers is 36 ÷ 12 = 3 layers. Choice A uses incorrect division (36 ÷ 18). Choice C confuses the length measurement with height. Choice D incorrectly divides 36 by the width only.

Since 64 unit cubes fill the container completely, filling it halfway up means using half the height, which requires 64 ÷ 2 = 32 cubes. Choice B incorrectly divides by 4 instead of 2. Choice C uses an arbitrary fraction. Choice D represents three-quarters full instead of half full.

When you encounter problems about finding the volume of irregular objects using displacement, think about what's actually being measured. The key insight is that the volume of the displaced material equals the volume of the object causing the displacement.

In this problem, Amy starts with 125 unit cubes in a 5×5×5 arrangement. When she places her irregular object in the container, she has to remove 18 unit cubes to make room for it. This means the object displaces exactly 18 unit cubes worth of space, so its volume is 18 cubic units.

Let's examine why the other answers are incorrect. Answer A (125 cubic units) represents the original volume of the container before the object was added - this doesn't tell us anything about the object's size. Answer B (107 cubic units) comes from subtracting the displaced cubes from the original amount (125 - 18 = 107), but this would represent how many cubes remain in the container, not the object's volume. Answer C (143 cubic units) adds the original cubes to the displaced amount (125 + 18 = 143), which doesn't represent any meaningful measurement in this context.

The correct answer is D (18 cubic units) because the object's volume equals exactly the amount of space it takes up - the 18 unit cubes that had to be removed.

Remember this key principle for displacement problems: the volume of the object always equals the volume of the material it displaces, not the original container volume or any combination involving it.

Volume can be measured by counting the unit cubes that make up a solid. To find the total volume, you need to count all the cubes, even if the layers are not the same size. You can use layers or rows by calculating the cubes in each layer separately—bottom layer 4 rows of 2 equals 8, top layer 2 rows of 2 equals 4—and adding them. This counting connects to total volume because it includes every cube in the structure for 12 cubic units. A common misconception is assuming all layers are full rectangles, but here the top is partial. In general, counting cubes helps measure volume by adapting to irregular shapes. This method ensures accuracy when layers differ in size.

Volume can be measured by counting the number of unit cubes that make up a solid. Count all visible cubes based on the views and depth provided. Use dimensions like width, height, and depth to systematize counting. Multiplying if rectangular or adding if irregular gives the volume. A misconception is assuming depth adds extra counts beyond the actual cubes, but count based on given depth. In general, counting cubes measures the solid's volume accurately. This approach generalizes to thin solids by accounting for all dimensions.

Volume can be measured by counting the number of unit cubes that make up a solid. To find the volume, you need to count all the unit cubes across all layers, making sure to include every one visible in the description. You can break it down by layers, counting the cubes in the bottom, middle, and top layers separately. The total volume is the sum of the cubes in each layer, resulting in the number of cubic units. One misconception is thinking that stacked layers mean you multiply the number of cubes per layer, but you simply add them since each cube is distinct. Counting unit cubes this way teaches us that volume represents the total amount of space the solid takes up. This approach generalizes to any stacked solid by adding up all individual cubes.