Question
Gina bought the small plastic boxes below.
A rectangular prism with a length of 5 centimeters, width of 4 centimeters, and height of 7 centimeters.A rectangular prism with a length of 4 centimeters, width of 3 centimeters, and height of 10 centimeters.
Which container requires more plastic to make, and how much more plastic is needed?
The container on the right is made of 2 centimeters squared more plastic.
The container on the left is made of 2 centimeters squared more plastic.
The container on the right is made of 20 centimeters squared more plastic.
The container on the left is made of 20 centimeters squared more plastic.
A rectangular prism with a length of 5 centimeters, width of 4 centimeters, and height of 7 centimeters.A rectangular prism with a length of 4 centimeters, width of 3 centimeters, and height of 10 centimeters.
Which container requires more plastic to make, and how much more plastic is needed?
The container on the right is made of 2 centimeters squared more plastic.
The container on the left is made of 2 centimeters squared more plastic.
The container on the right is made of 20 centimeters squared more plastic.
The container on the left is made of 20 centimeters squared more plastic.
Answers
GPT-4o mini
To determine which container requires more plastic, we need to calculate the surface area of each rectangular prism. The surface area \( A \) of a rectangular prism can be calculated using the formula:
\[
A = 2(lw + lh + wh)
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
### Container on the Left
For the first rectangular prism (length = 5 cm, width = 4 cm, height = 7 cm):
\[
A = 2(5 \cdot 4 + 5 \cdot 7 + 4 \cdot 7)
\]
Calculating each term:
\[
5 \cdot 4 = 20
\]
\[
5 \cdot 7 = 35
\]
\[
4 \cdot 7 = 28
\]
Now, sum these areas:
\[
20 + 35 + 28 = 83
\]
Now, multiply by 2:
\[
A = 2 \cdot 83 = 166 \text{ cm}^2
\]
### Container on the Right
For the second rectangular prism (length = 4 cm, width = 3 cm, height = 10 cm):
\[
A = 2(4 \cdot 3 + 4 \cdot 10 + 3 \cdot 10)
\]
Calculating each term:
\[
4 \cdot 3 = 12
\]
\[
4 \cdot 10 = 40
\]
\[
3 \cdot 10 = 30
\]
Now, sum these areas:
\[
12 + 40 + 30 = 82
\]
Now, multiply by 2:
\[
A = 2 \cdot 82 = 164 \text{ cm}^2
\]
### Comparing the Surface Areas
- Surface area of container on the left: \( 166 \) cm²
- Surface area of container on the right: \( 164 \) cm²
The container on the left requires more plastic:
To find out how much more plastic is needed:
\[
166 - 164 = 2 \text{ cm}^2
\]
### Conclusion
The container on the left is made of **2 centimeters squared more plastic**.
\[
A = 2(lw + lh + wh)
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
### Container on the Left
For the first rectangular prism (length = 5 cm, width = 4 cm, height = 7 cm):
\[
A = 2(5 \cdot 4 + 5 \cdot 7 + 4 \cdot 7)
\]
Calculating each term:
\[
5 \cdot 4 = 20
\]
\[
5 \cdot 7 = 35
\]
\[
4 \cdot 7 = 28
\]
Now, sum these areas:
\[
20 + 35 + 28 = 83
\]
Now, multiply by 2:
\[
A = 2 \cdot 83 = 166 \text{ cm}^2
\]
### Container on the Right
For the second rectangular prism (length = 4 cm, width = 3 cm, height = 10 cm):
\[
A = 2(4 \cdot 3 + 4 \cdot 10 + 3 \cdot 10)
\]
Calculating each term:
\[
4 \cdot 3 = 12
\]
\[
4 \cdot 10 = 40
\]
\[
3 \cdot 10 = 30
\]
Now, sum these areas:
\[
12 + 40 + 30 = 82
\]
Now, multiply by 2:
\[
A = 2 \cdot 82 = 164 \text{ cm}^2
\]
### Comparing the Surface Areas
- Surface area of container on the left: \( 166 \) cm²
- Surface area of container on the right: \( 164 \) cm²
The container on the left requires more plastic:
To find out how much more plastic is needed:
\[
166 - 164 = 2 \text{ cm}^2
\]
### Conclusion
The container on the left is made of **2 centimeters squared more plastic**.