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 8 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 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.

1 answer

To analyze the relationship between the surface areas of rectangular prisms A and B, we first note the given information:

  • Surface Area of Prism A = 25 square units
  • Surface Area of Prism B = 100 square units

We see that the surface area of prism B is 4 times that of prism A (100 / 25 = 4).

The surface area \( S \) of a rectangular prism is given by the formula: \[ S = 2(ab + ac + bc) \] where \( a \), \( b \), and \( c \) are the dimensions of the prism.

While the surface area increases with the dimensions, the relationship is not linear; instead, it's based on the square of the scaling factor of the dimensions. If the dimensions of prism A are multiplied by a factor \( k \), the surface area of prism B would be: \[ S_B = 2(k a)(k b) + 2(k a)(k c) + 2(k b)(k c) = k^2 \cdot S_A \]

Since we have determined that the surface area of prism B is \( 4 \) times that of prism A (i.e., \( k^2 = 4 \)), we can find \( k \) by taking the square root: \[ k = \sqrt{4} = 2 \]

This means that each dimension of prism B is 2 times that of prism A.

Thus, 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.