Rectangular prism A

Question

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)
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 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.

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 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.

GPT-4o mini · Answer

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, respectively. When comparing two prisms, if the surface area of prism B is $ k $ times that of prism A (in this case, $ k = 4 $, since 100 is 4 times 25), we cannot directly deduce that the dimensions of prism B are $ k $ times the dimensions of prism A. In fact, if the dimensions of prism A are scaled by a factor $ r $, the surface area scales by $ r^2 $. Thus, if the surface area of prism B is 4 times that of prism A, this suggests that the dimensions of prism B are scaled by a factor of: $$ r^2 = 4 \implies r = 2 $$ Therefore, the dimensions of prism B are twice as large as those 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."