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
Answers
GPT-4o mini
To compare the dimensions of rectangular prism A (with a surface area of 25 square units) and rectangular prism B (with a surface area of 100 square units), we first need to understand how surface area relates to the dimensions of rectangular prisms.
The surface area \( S \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by the formula:
\[
S = 2(lw + lh + wh)
\]
Since the surface area of prism B is four times that of prism A (100 square units compared to 25 square units), we can infer the following:
1. **Surface Area Growth**: The surface area of a prism scales with the square of the scaling factor of its linear dimensions. If the surface area of prism B is four times that of prism A, that suggests a scaling factor for the linear dimensions of prism B compared to prism A.
2. **Scaling Factor**: The scaling factor \( k \) in the dimensions can be found by solving the equation:
\[
k^2 = \frac{\text{Surface Area of B}}{\text{Surface Area of A}} = \frac{100}{25} = 4
\]
Thus, \( k = 2 \).
This means that each linear dimension of prism B (length, width, and height) is 2 times that of prism A.
**Conclusion**: The dimensions of rectangular prism B are 2 times greater than the dimensions of rectangular prism A. Thus, if you know the dimensions (length, width, height) of prism A, you can find those of prism B by multiplying each by 2.
The surface area \( S \) of a rectangular prism with dimensions \( l \), \( w \), and \( h \) is given by the formula:
\[
S = 2(lw + lh + wh)
\]
Since the surface area of prism B is four times that of prism A (100 square units compared to 25 square units), we can infer the following:
1. **Surface Area Growth**: The surface area of a prism scales with the square of the scaling factor of its linear dimensions. If the surface area of prism B is four times that of prism A, that suggests a scaling factor for the linear dimensions of prism B compared to prism A.
2. **Scaling Factor**: The scaling factor \( k \) in the dimensions can be found by solving the equation:
\[
k^2 = \frac{\text{Surface Area of B}}{\text{Surface Area of A}} = \frac{100}{25} = 4
\]
Thus, \( k = 2 \).
This means that each linear dimension of prism B (length, width, and height) is 2 times that of prism A.
**Conclusion**: The dimensions of rectangular prism B are 2 times greater than the dimensions of rectangular prism A. Thus, if you know the dimensions (length, width, height) of prism A, you can find those of prism B by multiplying each by 2.