Question
The cracker manufacturer is trying to determine the best dimensions for the boxes to hold their product so that they use the least amount of cardboard.
A rectangular prism with length of 6 inches, width of 4 inches, and height of 12 inches.A rectangular prism with length of 7 inches, width of 6 inches, and height of 9 inches.
Which box uses less cardboard, and how much less cardboard does it use?
The box on the right uses 30 inches squared less cardboard.
The box on the left uses 30 inches squared less cardboard.
The box on the right uses 90 inches squared less cardboard.
The box on the left uses 90 inches squared less cardboard.
A rectangular prism with length of 6 inches, width of 4 inches, and height of 12 inches.A rectangular prism with length of 7 inches, width of 6 inches, and height of 9 inches.
Which box uses less cardboard, and how much less cardboard does it use?
The box on the right uses 30 inches squared less cardboard.
The box on the left uses 30 inches squared less cardboard.
The box on the right uses 90 inches squared less cardboard.
The box on the left uses 90 inches squared less cardboard.
Answers
GPT-4o mini
To determine which box uses less cardboard, we need to calculate the surface area of both rectangular prisms. 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 of the prism.
**For the first box (length = 6 inches, width = 4 inches, height = 12 inches):**
\[
A_1 = 2(6 \cdot 4 + 6 \cdot 12 + 4 \cdot 12)
\]
Calculating each term:
- \(6 \cdot 4 = 24\)
- \(6 \cdot 12 = 72\)
- \(4 \cdot 12 = 48\)
Adding these together:
\[
A_1 = 2(24 + 72 + 48) = 2(144) = 288 \text{ square inches}
\]
**For the second box (length = 7 inches, width = 6 inches, height = 9 inches):**
\[
A_2 = 2(7 \cdot 6 + 7 \cdot 9 + 6 \cdot 9)
\]
Calculating each term:
- \(7 \cdot 6 = 42\)
- \(7 \cdot 9 = 63\)
- \(6 \cdot 9 = 54\)
Adding these together:
\[
A_2 = 2(42 + 63 + 54) = 2(159) = 318 \text{ square inches}
\]
Now, comparing the surface areas:
- Surface area of the first box: \(A_1 = 288\) square inches
- Surface area of the second box: \(A_2 = 318\) square inches
To find out how much less cardboard the first box uses compared to the second box:
\[
A_2 - A_1 = 318 - 288 = 30 \text{ square inches}
\]
Thus, the first box (left box) uses 30 square inches less cardboard than the second box (right box).
The correct answer is: **The box on the left uses 30 inches squared less cardboard.**
\[
A = 2(lw + lh + wh)
\]
Where \(l\) is the length, \(w\) is the width, and \(h\) is the height of the prism.
**For the first box (length = 6 inches, width = 4 inches, height = 12 inches):**
\[
A_1 = 2(6 \cdot 4 + 6 \cdot 12 + 4 \cdot 12)
\]
Calculating each term:
- \(6 \cdot 4 = 24\)
- \(6 \cdot 12 = 72\)
- \(4 \cdot 12 = 48\)
Adding these together:
\[
A_1 = 2(24 + 72 + 48) = 2(144) = 288 \text{ square inches}
\]
**For the second box (length = 7 inches, width = 6 inches, height = 9 inches):**
\[
A_2 = 2(7 \cdot 6 + 7 \cdot 9 + 6 \cdot 9)
\]
Calculating each term:
- \(7 \cdot 6 = 42\)
- \(7 \cdot 9 = 63\)
- \(6 \cdot 9 = 54\)
Adding these together:
\[
A_2 = 2(42 + 63 + 54) = 2(159) = 318 \text{ square inches}
\]
Now, comparing the surface areas:
- Surface area of the first box: \(A_1 = 288\) square inches
- Surface area of the second box: \(A_2 = 318\) square inches
To find out how much less cardboard the first box uses compared to the second box:
\[
A_2 - A_1 = 318 - 288 = 30 \text{ square inches}
\]
Thus, the first box (left box) uses 30 square inches less cardboard than the second box (right box).
The correct answer is: **The box on the left uses 30 inches squared less cardboard.**