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 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.

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.

1 answer

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 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, then the surface area of prism B is 4 times that of prism A:

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

When the surface area of a rectangular prism scales by a factor of \( k \), the dimensions scale by the cube root of that factor. Therefore:

\[ \text{If } S_B = 4S_A, \text{ then } \text{dimensions of prism B} = \sqrt[3]{4} \times \text{dimensions of prism A} \]

Calculating \( \sqrt[3]{4} \):

\[ \sqrt[3]{4} \approx 1.5874 \]

This means that the dimensions of prism B are approximately 1.59 times those of prism A, which does not correlate to doubling.

Among the given statements, the correct comparison 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."

However, it's important to note that this statement is mathematically incorrect in the context of surface area and dimension scaling, so none of the responses provided are accurate based on the mathematical principles involved.

To conclude, correct reasoning should state that the dimensions of prism B are the cube root of 4 times larger than those of prism A, not double or any other simple integer relation.