To determine the scale factor between two similar hexagonal prisms, we can use the relationship between their surface areas. If the surface areas of two similar solids are known, the scale factor between their corresponding dimensions can be found by taking the square root of the ratio of their areas.
Given:
- The surface area of the smaller hexagonal prism: [tex]\( A_{\text{smaller}} = 882 \)[/tex] cm²
- The surface area of the larger hexagonal prism: [tex]\( A_{\text{larger}} = 1058 \)[/tex] cm²
The formula for the scale factor [tex]\( k \)[/tex] relating the two prisms' dimensions is:
[tex]\[ k = \sqrt{\frac{A_{\text{smaller}}}{A_{\text{larger}}}} \][/tex]
First, calculate the ratio of their surface areas:
[tex]\[ \frac{882}{1058} \approx 0.8336 \][/tex]
Next, take the square root of this ratio to find the scale factor:
[tex]\[ k = \sqrt{0.8336} \approx 0.9130 \][/tex]
Now, we need to compare this calculated scale factor to the choices given:
A. [tex]\( \frac{882}{1058} \approx 0.8336 \)[/tex]
B. [tex]\( \frac{21}{23} \approx 0.9130 \)[/tex]
C. [tex]\( \frac{23}{21} \approx 1.0952 \)[/tex]
D. [tex]\( \frac{21}{11} \approx 1.9091 \)[/tex]
From this comparison, it is clear that choice B, [tex]\( \frac{21}{23} \)[/tex], most closely matches the calculated scale factor of approximately 0.9130.
Thus, the best answer is:
B. [tex]\( \frac{21}{23} \)[/tex]
