Rectangular prism A

Question

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)
Responses

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 upper B ’s surface area is 4 times as large as prism upper A ’s surface area, prism upper B ’s dimensions are 8 times as large as prism upper 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 upper B ’s surface area is 4 times as large as prism upper A ’s surface area, prism upper B ’s dimensions are twice as large as prism upper 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 upper B ’s surface area is 4 times as large as prism upper A ’s surface area, prism upper B ’s dimensions are half as large as prism upper 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 upper B ’s surface area is 4 times as large as prism upper A ’s surface area, prism upper B ’s dimensions are 16 times as large as prism upper A ’s.

Answer 1

The surface area \( S \) of a rectangular prism is given by the formula:

\[ S = 2(lw + lh + wh) \]

where \( l \), \( w \), and \( h \) are the dimensions (length, width, and height) of the prism.

If the surface area of prism \( A \) is \( 25 \) square units and the surface area of prism \( B \) is \( 100 \) square units, we can see that:

\[ \frac{S_B}{S_A} = \frac{100}{25} = 4 \]

This means prism \( B \) has a surface area that is \( 4 \) times larger than that of prism \( A \).

However, surface area scales with the square of the linear dimensions. If the dimensions of prism \( A \) are changed by a factor \( k \), then the surface area will change by a factor of \( k^2 \). Therefore:

\[ k^2 = 4 \implies k = 2 \]

This means the dimensions of prism \( B \) are \( 2 \) times as large as the dimensions 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.