To compare the dimensions of the two rectangular prisms based on their surface areas, we can use the relationship between the surface area and the dimensions of a rectangular prism.
The surface area \( S \) of a rectangular prism can be expressed as: \[ S = 2(lw + lh + wh) \] where \( l \), \( w \), and \( h \) are the length, width, and height of the prism.
If the dimensions of prism A are \( l_A \), \( w_A \), and \( h_A \), then its surface area is given by: \[ S_A = 2(l_A w_A + l_A h_A + w_A h_A) = 25 \text{ square units} \] If the dimensions of prism B are \( l_B \), \( w_B \), and \( h_B \), then its surface area is given by: \[ S_B = 2(l_B w_B + l_B h_B + w_B h_B) = 100 \text{ square units} \]
We know that the surface area of prism B is 4 times that of prism A: \[ S_B = 4 \cdot S_A \] This means: \[ 100 = 4 \cdot 25 \]
In general, when the surface area of a three-dimensional object increases by a factor, the linear dimensions (length, width, height) increase by the square root of that factor. Since the surface area of prism B is 4 times that of prism A, the relationship between their dimensions can be described as follows: \[ \text{Dimension factor} = \sqrt{4} = 2 \]
This means that each dimension of prism B is twice the corresponding dimension of prism A.
Therefore, the correct statement 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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