Question
Which would neatly fill the gap in the prism shown below? Check all that apply.
A prism has a length of 2 and one-half, height of 3, and width of 2. 12 blocks are inside of the prism.
6 blocks each measuring One-fifth times 1 times 1
12 blocks each measuring One-fourth times 1 times 1
6 blocks each measuring One-third times 1 times 1
6 blocks each measuring One-half times 1 times 1
2 blocks each measuring 1 times 1 times 1
A prism has a length of 2 and one-half, height of 3, and width of 2. 12 blocks are inside of the prism.
6 blocks each measuring One-fifth times 1 times 1
12 blocks each measuring One-fourth times 1 times 1
6 blocks each measuring One-third times 1 times 1
6 blocks each measuring One-half times 1 times 1
2 blocks each measuring 1 times 1 times 1
Answers
GPT-4o mini
To determine which blocks can neatly fill the gap in the prism, we first need to calculate the volume of the prism.
The prism has dimensions:
- Length = 2.5 units
- Height = 3 units
- Width = 2 units
The volume \( V \) of the prism can be calculated using the formula:
\[
V = \text{Length} \times \text{Height} \times \text{Width}
\]
Substituting in the values:
\[
V = 2.5 \times 3 \times 2 = 15 \text{ cubic units}
\]
Currently, there are 12 blocks inside the prism. To find the volume occupied by these blocks, we'll need to understand the volume of the blocks listed.
1. **6 blocks each measuring One-fifth times 1 times 1**:
- Volume of one block = \( \frac{1}{5} \times 1 \times 1 = \frac{1}{5} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{5} = \frac{6}{5} \)
2. **12 blocks each measuring One-fourth times 1 times 1**:
- Volume of one block = \( \frac{1}{4} \times 1 \times 1 = \frac{1}{4} \)
- Volume of 12 blocks = \( 12 \times \frac{1}{4} = 3 \)
3. **6 blocks each measuring One-third times 1 times 1**:
- Volume of one block = \( \frac{1}{3} \times 1 \times 1 = \frac{1}{3} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{3} = 2 \)
4. **6 blocks each measuring One-half times 1 times 1**:
- Volume of one block = \( \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{2} = 3 \)
5. **2 blocks each measuring 1 times 1 times 1**:
- Volume of one block = \( 1 \times 1 \times 1 = 1 \)
- Volume of 2 blocks = \( 2 \times 1 = 2 \)
Next, we sum the total volume of the blocks inside the prism:
\[
\text{Total volume of blocks} = \frac{6}{5} + 3 + 2 + 3 + 2
\]
Calculating:
\[
\frac{6}{5} + 3 + 2 + 3 + 2 = \frac{6}{5} + \frac{15}{5} + \frac{10}{5} + \frac{15}{5} + \frac{10}{5} = \frac{56}{5} = 11.2 \text{ cubic units}
\]
Since the total volume of the prism is 15 cubic units and the current total volume occupied by blocks is 11.2 cubic units, the gap in the prism is:
\[
15 - 11.2 = 3.8 \text{ cubic units}
\]
Now, we can check which blocks would neatly fill the remaining volume of 3.8 cubic units:
1. **6 blocks each measuring One-fifth times 1 times 1**:
- Volume = \( \frac{1}{5} \times 6 = \frac{6}{5} = 1.2 \) (can be used)
2. **12 blocks each measuring One-fourth times 1 times 1**:
- Volume = \( \frac{1}{4} \times 12 = 3 \) (can be used)
3. **6 blocks each measuring One-third times 1 times 1**:
- Volume = \( \frac{1}{3} \times 6 = 2 \) (can be used)
4. **6 blocks each measuring One-half times 1 times 1**:
- Volume = \( \frac{1}{2} \times 6 = 3 \) (can be used)
5. **2 blocks each measuring 1 times 1 times 1**:
- Volume = \( 1 \times 2 = 2 \) (can be used)
To fill the gap of 3.8 cubic units, any combination of the blocks that adds up to 3.8 is acceptable.
Based on the volumes calculated, the following individual block types can be selected to fill the gap:
- **12 blocks each measuring One-fourth times 1 times 1** (3 cubic units)
- **6 blocks each measuring One-half times 1 times 1** (3 cubic units)
- **6 blocks each measuring One-third times 1 times 1** (2 cubic units) combined with part of another block type to achieve 3.8
Thus, the block types that can help fill the gap are:
- 12 blocks each measuring One-fourth times 1 times 1
- 6 blocks each measuring One-half times 1 times 1
- 6 blocks each measuring One-third times 1 times 1
So, the selections that apply would be:
- 12 blocks each measuring One-fourth times 1 times 1
- 6 blocks each measuring One-half times 1 times 1
- 6 blocks each measuring One-third times 1 times 1
These combinations would fit to fill the remaining volume in the prism neatly.
The prism has dimensions:
- Length = 2.5 units
- Height = 3 units
- Width = 2 units
The volume \( V \) of the prism can be calculated using the formula:
\[
V = \text{Length} \times \text{Height} \times \text{Width}
\]
Substituting in the values:
\[
V = 2.5 \times 3 \times 2 = 15 \text{ cubic units}
\]
Currently, there are 12 blocks inside the prism. To find the volume occupied by these blocks, we'll need to understand the volume of the blocks listed.
1. **6 blocks each measuring One-fifth times 1 times 1**:
- Volume of one block = \( \frac{1}{5} \times 1 \times 1 = \frac{1}{5} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{5} = \frac{6}{5} \)
2. **12 blocks each measuring One-fourth times 1 times 1**:
- Volume of one block = \( \frac{1}{4} \times 1 \times 1 = \frac{1}{4} \)
- Volume of 12 blocks = \( 12 \times \frac{1}{4} = 3 \)
3. **6 blocks each measuring One-third times 1 times 1**:
- Volume of one block = \( \frac{1}{3} \times 1 \times 1 = \frac{1}{3} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{3} = 2 \)
4. **6 blocks each measuring One-half times 1 times 1**:
- Volume of one block = \( \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \)
- Volume of 6 blocks = \( 6 \times \frac{1}{2} = 3 \)
5. **2 blocks each measuring 1 times 1 times 1**:
- Volume of one block = \( 1 \times 1 \times 1 = 1 \)
- Volume of 2 blocks = \( 2 \times 1 = 2 \)
Next, we sum the total volume of the blocks inside the prism:
\[
\text{Total volume of blocks} = \frac{6}{5} + 3 + 2 + 3 + 2
\]
Calculating:
\[
\frac{6}{5} + 3 + 2 + 3 + 2 = \frac{6}{5} + \frac{15}{5} + \frac{10}{5} + \frac{15}{5} + \frac{10}{5} = \frac{56}{5} = 11.2 \text{ cubic units}
\]
Since the total volume of the prism is 15 cubic units and the current total volume occupied by blocks is 11.2 cubic units, the gap in the prism is:
\[
15 - 11.2 = 3.8 \text{ cubic units}
\]
Now, we can check which blocks would neatly fill the remaining volume of 3.8 cubic units:
1. **6 blocks each measuring One-fifth times 1 times 1**:
- Volume = \( \frac{1}{5} \times 6 = \frac{6}{5} = 1.2 \) (can be used)
2. **12 blocks each measuring One-fourth times 1 times 1**:
- Volume = \( \frac{1}{4} \times 12 = 3 \) (can be used)
3. **6 blocks each measuring One-third times 1 times 1**:
- Volume = \( \frac{1}{3} \times 6 = 2 \) (can be used)
4. **6 blocks each measuring One-half times 1 times 1**:
- Volume = \( \frac{1}{2} \times 6 = 3 \) (can be used)
5. **2 blocks each measuring 1 times 1 times 1**:
- Volume = \( 1 \times 2 = 2 \) (can be used)
To fill the gap of 3.8 cubic units, any combination of the blocks that adds up to 3.8 is acceptable.
Based on the volumes calculated, the following individual block types can be selected to fill the gap:
- **12 blocks each measuring One-fourth times 1 times 1** (3 cubic units)
- **6 blocks each measuring One-half times 1 times 1** (3 cubic units)
- **6 blocks each measuring One-third times 1 times 1** (2 cubic units) combined with part of another block type to achieve 3.8
Thus, the block types that can help fill the gap are:
- 12 blocks each measuring One-fourth times 1 times 1
- 6 blocks each measuring One-half times 1 times 1
- 6 blocks each measuring One-third times 1 times 1
So, the selections that apply would be:
- 12 blocks each measuring One-fourth times 1 times 1
- 6 blocks each measuring One-half times 1 times 1
- 6 blocks each measuring One-third times 1 times 1
These combinations would fit to fill the remaining volume in the prism neatly.