To analyze the comparison of the dimensions of the two rectangular prisms based on their surface areas, we need to understand how surface area, volume, and dimensions are related.
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 prism B has a surface area that is 4 times that of prism A (since \( 100 / 25 = 4 \)), we can reason about how this affects the dimensions.
When the dimensions of a prism are all scaled by a factor \( k \), the surface area scales by \( k^2 \). Therefore, if the surface area of prism B is 4 times that of prism A, we have:
\[ \frac{S_B}{S_A} = \frac{100}{25} = 4 = k^2 \]
To find \( k \), we take the square root:
\[ k = \sqrt{4} = 2 \]
This means that each dimension of prism B is 2 times that of prism A.
Now, let's look at the statements provided:
A. 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. (Incorrect, they are 2 times, not 16)
B. 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. (Incorrect, they are 2 times, not 8)
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. (Correct)
D. 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. (Incorrect, they are larger, not half)
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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