Question
Use unit cubes to find the volume of 9 right rectangular prisms if each prism has width 3 in., length 5 in., and height 2 in. Then use a formula to find the volume of the 9 prisms.
Use pencil and paper. Explain which method you prefer and why.
Use pencil and paper. Explain which method you prefer and why.
Answers
GPT-4o mini
To find the volume of a right rectangular prism, you can use unit cubes or a mathematical formula. Let's solve this both ways.
### 1. Using Unit Cubes Method
To visualize the volume of one right rectangular prism using unit cubes, imagine that each prism can be filled with 1-inch cubes.
#### Dimensions of One Prism:
- Width: 3 in (3 cubes wide)
- Length: 5 in (5 cubes long)
- Height: 2 in (2 cubes high)
You can fill the prism layer by layer:
- Each layer (slice of height 1 inch) contains a base area of width × length.
- The base area of one layer is:
\[
\text{Base Area} = \text{Width} \times \text{Length} = 3 \, \text{in} \times 5 \, \text{in} = 15 \, \text{square inches}
\]
Since the height of the prism is 2 inches (2 layers):
- Total volume of one prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 15 \, \text{square inches} \times 2 \, \text{inches} = 30 \, \text{cubic inches}
\]
For 9 such prisms, the total volume using the unit cubes method:
\[
\text{Total Volume} = 9 \times 30 \, \text{cubic inches} = 270 \, \text{cubic inches}
\]
### 2. Using the Formula Method
The formula for the volume \( V \) of a right rectangular prism is given by:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Plugging in the dimensions of one prism:
\[
V = 5 \, \text{inches} \times 3 \, \text{inches} \times 2 \, \text{inches} = 30 \, \text{cubic inches}
\]
Now, for 9 prisms, we multiply by 9:
\[
\text{Total Volume} = 9 \times 30 \, \text{cubic inches} = 270 \, \text{cubic inches}
\]
### Conclusion
Both methods yield the same total volume of 270 cubic inches for the 9 prisms.
### Preference
**Preferred Method: Formula Method**
I prefer using the formula method for a few reasons:
1. **Efficiency**: It is quicker since it eliminates the need to visually represent and count unit cubes.
2. **Simplicity**: The formula is straightforward and looks cleaner, which is especially useful for larger calculations or when working with more complex dimensions.
3. **Generalization**: The formula can easily be applied to prisms of any dimensions without needing to create a physical model or visualize the unit cubes.
In educational settings, using unit cubes can be very helpful for building intuition about volumes, especially for visual learners. However, for practical applications, the formula method provides a more efficient approach.
### 1. Using Unit Cubes Method
To visualize the volume of one right rectangular prism using unit cubes, imagine that each prism can be filled with 1-inch cubes.
#### Dimensions of One Prism:
- Width: 3 in (3 cubes wide)
- Length: 5 in (5 cubes long)
- Height: 2 in (2 cubes high)
You can fill the prism layer by layer:
- Each layer (slice of height 1 inch) contains a base area of width × length.
- The base area of one layer is:
\[
\text{Base Area} = \text{Width} \times \text{Length} = 3 \, \text{in} \times 5 \, \text{in} = 15 \, \text{square inches}
\]
Since the height of the prism is 2 inches (2 layers):
- Total volume of one prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 15 \, \text{square inches} \times 2 \, \text{inches} = 30 \, \text{cubic inches}
\]
For 9 such prisms, the total volume using the unit cubes method:
\[
\text{Total Volume} = 9 \times 30 \, \text{cubic inches} = 270 \, \text{cubic inches}
\]
### 2. Using the Formula Method
The formula for the volume \( V \) of a right rectangular prism is given by:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Plugging in the dimensions of one prism:
\[
V = 5 \, \text{inches} \times 3 \, \text{inches} \times 2 \, \text{inches} = 30 \, \text{cubic inches}
\]
Now, for 9 prisms, we multiply by 9:
\[
\text{Total Volume} = 9 \times 30 \, \text{cubic inches} = 270 \, \text{cubic inches}
\]
### Conclusion
Both methods yield the same total volume of 270 cubic inches for the 9 prisms.
### Preference
**Preferred Method: Formula Method**
I prefer using the formula method for a few reasons:
1. **Efficiency**: It is quicker since it eliminates the need to visually represent and count unit cubes.
2. **Simplicity**: The formula is straightforward and looks cleaner, which is especially useful for larger calculations or when working with more complex dimensions.
3. **Generalization**: The formula can easily be applied to prisms of any dimensions without needing to create a physical model or visualize the unit cubes.
In educational settings, using unit cubes can be very helpful for building intuition about volumes, especially for visual learners. However, for practical applications, the formula method provides a more efficient approach.