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Rectangular prism A has a surface area of 25 square units and rectangular prism B has a surface area of 100 square units. Which...Question
Rectangular prism A has a surface area of 25 square units and rectangular prism
B has a surface area of 100 square units. Which statement accurately compares the dimensions of prism A to those of prism B? (1 point)
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are half as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are twice as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are 16 times as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are 8 times as large as prism A's.
B has a surface area of 100 square units. Which statement accurately compares the dimensions of prism A to those of prism B? (1 point)
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are half as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are twice as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are 16 times as large as prism A's.
Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are 8 times as large as prism A's.
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Answered by
GPT-4o mini
To compare the dimensions of rectangular prisms A and B based on their surface areas, we need to understand how surface area relates to dimensions.
The surface area \( S \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by the formula:
\[
S = 2(lw + lh + wh)
\]
If the surface area of prism B is 4 times that of prism A, we can represent this mathematically as:
\[
S_B = 4S_A
\]
Given:
- \( S_A = 25 \) square units
- \( S_B = 100 \) square units
From the given information, it's clear that the surface area of prism B is indeed 4 times that of prism A (100 = 4 * 25).
However, to determine how the dimensions relate to each other, we need to analyze the scaling factor.
If we denote the dimensions of prism A as \( (l, w, h) \), then the dimensions of prism B can be represented as \( (k \cdot l, k \cdot w, k \cdot h) \) for some scaling factor \( k \). The new surface area would then be:
\[
S_B = 2(k \cdot l)(k \cdot w) + 2(k \cdot l)(k \cdot h) + 2(k \cdot w)(k \cdot h)
\]
This can be simplified to:
\[
S_B = k^2 \cdot S_A
\]
Given that \( S_B = 4S_A \), we set up the equation:
\[
k^2 \cdot S_A = 4S_A
\]
Assuming \( S_A \) is not zero, we can divide both sides by \( S_A \):
\[
k^2 = 4
\]
Taking the square root:
\[
k = 2
\]
This shows that each dimension of prism B is actually twice the dimension of prism A.
Thus, the correct statement comparing the dimensions of prism A to those of prism B is:
**Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are twice as large as prism A's.**
The surface area \( S \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by the formula:
\[
S = 2(lw + lh + wh)
\]
If the surface area of prism B is 4 times that of prism A, we can represent this mathematically as:
\[
S_B = 4S_A
\]
Given:
- \( S_A = 25 \) square units
- \( S_B = 100 \) square units
From the given information, it's clear that the surface area of prism B is indeed 4 times that of prism A (100 = 4 * 25).
However, to determine how the dimensions relate to each other, we need to analyze the scaling factor.
If we denote the dimensions of prism A as \( (l, w, h) \), then the dimensions of prism B can be represented as \( (k \cdot l, k \cdot w, k \cdot h) \) for some scaling factor \( k \). The new surface area would then be:
\[
S_B = 2(k \cdot l)(k \cdot w) + 2(k \cdot l)(k \cdot h) + 2(k \cdot w)(k \cdot h)
\]
This can be simplified to:
\[
S_B = k^2 \cdot S_A
\]
Given that \( S_B = 4S_A \), we set up the equation:
\[
k^2 \cdot S_A = 4S_A
\]
Assuming \( S_A \) is not zero, we can divide both sides by \( S_A \):
\[
k^2 = 4
\]
Taking the square root:
\[
k = 2
\]
This shows that each dimension of prism B is actually twice the dimension of prism A.
Thus, the correct statement comparing the dimensions of prism A to those of prism B is:
**Since prism B's surface area is 4 times as large as prism A's surface area, prism B's dimensions are twice as large as prism A's.**
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