To solve the problem of determining the scale factor between two similar triangular prisms with volumes of [tex]\(512 \, \text{ft}^3\)[/tex] and [tex]\(1331 \, \text{ft}^3\)[/tex], follow these steps:
1. Understand the relation between volumes and scale factors:
- The volumes of similar shapes are related by the cube of the scale factor. If the scale factor is [tex]\(k\)[/tex], then the volume ratio is [tex]\(k^3\)[/tex].
2. Calculate the volume ratio:
- Volume of the first prism: [tex]\(512 \, \text{ft}^3\)[/tex]
- Volume of the second prism: [tex]\(1331 \, \text{ft}^3\)[/tex]
- Volume ratio: [tex]\(\frac{\text{Volume of the first}}{\text{Volume of the second}} = \frac{512}{1331}\)[/tex]
3. Determine the scale factor:
- Since [tex]\( \left(\text{scale factor}\right)^3 = \frac{512}{1331} \)[/tex], you need to find the cube root of [tex]\(\frac{512}{1331}\)[/tex]:
[tex]\[
\text{scale factor} = \left(\frac{512}{1331}\right)^{1/3}
\][/tex]
4. Compute the approximate value of the scale factor:
[tex]\[
\left(\frac{512}{1331}\right)^{1/3} \approx 0.727
\][/tex]
5. Compare the computed scale factor to the given choices:
- Choice A: [tex]\(\frac{64}{121} \approx 0.529\)[/tex]
- Choice B: [tex]\(\frac{8}{11} \approx 0.727\)[/tex]
- Choice C: [tex]\(\frac{4}{5} = 0.8\)[/tex]
- Choice D: [tex]\(\frac{11}{12} \approx 0.917\)[/tex]
6. Identify the best match:
- The scale factor [tex]\(0.727\)[/tex] closely matches [tex]\(\frac{8}{11}\)[/tex] from Choice B.
Thus, the best answer is:
[tex]\[
\boxed{B \, \text{(} \frac{8}{11} \text{)}}
\][/tex]
