To understand the relationship between the surface areas and the dimensions of the rectangular prisms A and B, we need to look at how surface area affects dimensions geometrically.
The surface area \(S\) of a rectangular prism with dimensions \(a\), \(b\), and \(c\) is given by the formula:
\[ S = 2(ab + ac + bc) \]
If the dimensions of prism B are scaled by a factor \(k\) compared to prism A (i.e., if the dimensions of prism A are \(a\), \(b\), and \(c\), then the dimensions of prism B are \(ka\), \(kb\), and \(kc\)), the surface area of prism B becomes:
\[ S_B = 2(k^2ab + k^2ac + k^2bc) = k^2 \times 2(ab + ac + bc) = k^2 S_A \]
Given that the surface area of prism B is 4 times that of prism A (\(S_B = 4S_A\)), we can equate the surface area expressions:
\[ k^2 S_A = 4 S_A \]
Assuming \(S_A \neq 0\), we can divide both sides by \(S_A\):
\[ k^2 = 4 \]
Taking the positive square root, we find:
\[ k = 2 \]
This means that the dimensions of prism B are twice as large as those of prism A.
Therefore, the correct statement is:
c. 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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