Question
How many one-third cubes are needed to fill the gap in the prism shown below?
A prism has a length of 4 and two-thirds, height of 2, and width of 4. 16 blocks are inside of the prism.
4
8
16
24
A prism has a length of 4 and two-thirds, height of 2, and width of 4. 16 blocks are inside of the prism.
4
8
16
24
Answers
GPT-4o mini
To find out how many one-third cubes are needed to fill the gap in the prism, we first need to calculate the volume of the prism and then determine how many one-third cubes would be required to fill that volume.
1. **Calculate the volume of the prism**:
- The dimensions of the prism are length = \(4 \frac{2}{3}\), height = \(2\), and width = \(4\).
- Convert \(4 \frac{2}{3}\) to an improper fraction:
\[
4 \frac{2}{3} = \frac{14}{3}
\]
- The volume \(V\) of the prism is given by:
\[
V = \text{length} \times \text{height} \times \text{width} = \left(\frac{14}{3}\right) \times 2 \times 4
\]
- Calculate step by step:
\[
2 \times 4 = 8
\]
\[
V = \left(\frac{14}{3}\right) \times 8 = \frac{14 \times 8}{3} = \frac{112}{3}
\]
2. **Calculate the volume of the blocks inside the prism**:
- We know there are 16 blocks inside the prism. Since we don't know the volume of these blocks, we assume they are standard cubes with volume \(1\) each.
- Thus, the total volume occupied by the blocks is:
\[
\text{Volume of blocks} = 16 \times 1 = 16
\]
3. **Calculate the volume of the gap**:
- To find the gap, we subtract the volume of the blocks from the volume of the prism:
\[
\text{Volume of gap} = V - \text{Volume of blocks} = \frac{112}{3} - 16
\]
- Convert \(16\) to a fraction that has a common denominator with \(\frac{112}{3}\):
\[
16 = \frac{48}{3}
\]
- Now perform the subtraction:
\[
\text{Volume of gap} = \frac{112}{3} - \frac{48}{3} = \frac{64}{3}
\]
4. **Calculate how many one-third cubes fill the gap**:
- The volume of a one-third cube is:
\[
\text{Volume of one-third cube} = \left(\frac{1}{3}\right)^3 = \frac{1}{27}
\]
- To find out how many one-third cubes are needed to fill the volume of the gap \(\frac{64}{3}\):
\[
\text{Number of cubes} = \frac{\text{Volume of gap}}{\text{Volume of one-third cube}} = \frac{\frac{64}{3}}{\frac{1}{27}} = \frac{64}{3} \times 27 = \frac{64 \times 27}{3} = \frac{1728}{3} = 576
\]
Finally, the number of one-third cubes needed to fill the gap in the prism is \(576\). Given the choices provided (4, 8, 16, 24), this suggests there may be a discrepancy as this number is much larger. However, the calculation reveals the required number based on the provided dimensions and assumptions surrounding cube volumes. Please check the dimensions again for further accuracy or constraints on sizes.
1. **Calculate the volume of the prism**:
- The dimensions of the prism are length = \(4 \frac{2}{3}\), height = \(2\), and width = \(4\).
- Convert \(4 \frac{2}{3}\) to an improper fraction:
\[
4 \frac{2}{3} = \frac{14}{3}
\]
- The volume \(V\) of the prism is given by:
\[
V = \text{length} \times \text{height} \times \text{width} = \left(\frac{14}{3}\right) \times 2 \times 4
\]
- Calculate step by step:
\[
2 \times 4 = 8
\]
\[
V = \left(\frac{14}{3}\right) \times 8 = \frac{14 \times 8}{3} = \frac{112}{3}
\]
2. **Calculate the volume of the blocks inside the prism**:
- We know there are 16 blocks inside the prism. Since we don't know the volume of these blocks, we assume they are standard cubes with volume \(1\) each.
- Thus, the total volume occupied by the blocks is:
\[
\text{Volume of blocks} = 16 \times 1 = 16
\]
3. **Calculate the volume of the gap**:
- To find the gap, we subtract the volume of the blocks from the volume of the prism:
\[
\text{Volume of gap} = V - \text{Volume of blocks} = \frac{112}{3} - 16
\]
- Convert \(16\) to a fraction that has a common denominator with \(\frac{112}{3}\):
\[
16 = \frac{48}{3}
\]
- Now perform the subtraction:
\[
\text{Volume of gap} = \frac{112}{3} - \frac{48}{3} = \frac{64}{3}
\]
4. **Calculate how many one-third cubes fill the gap**:
- The volume of a one-third cube is:
\[
\text{Volume of one-third cube} = \left(\frac{1}{3}\right)^3 = \frac{1}{27}
\]
- To find out how many one-third cubes are needed to fill the volume of the gap \(\frac{64}{3}\):
\[
\text{Number of cubes} = \frac{\text{Volume of gap}}{\text{Volume of one-third cube}} = \frac{\frac{64}{3}}{\frac{1}{27}} = \frac{64}{3} \times 27 = \frac{64 \times 27}{3} = \frac{1728}{3} = 576
\]
Finally, the number of one-third cubes needed to fill the gap in the prism is \(576\). Given the choices provided (4, 8, 16, 24), this suggests there may be a discrepancy as this number is much larger. However, the calculation reveals the required number based on the provided dimensions and assumptions surrounding cube volumes. Please check the dimensions again for further accuracy or constraints on sizes.