Let’s determine which of the given prisms is similar to the prism with dimensions [tex]\( \text{length} = 5.2 \)[/tex] units, [tex]\( \text{width} = 2 \)[/tex] units, and [tex]\( \text{height} = 3 \)[/tex] units.
To establish similarity between two right rectangular prisms, their corresponding dimensions should be proportional. We need to find the ratios of dimensions (length to width, width to height, and length to height) and compare these ratios among the prisms.
For the given prism:
- Length-to-width ratio [tex]\( = \frac{5.2}{2} = 2.6 \)[/tex]
- Width-to-height ratio [tex]\( = \frac{2}{3} \approx 0.67 \)[/tex]
- Length-to-height ratio [tex]\( = \frac{5.2}{3} \approx 1.73 \)[/tex]
Prism 1:
- Dimensions: [tex]\( \text{length} = 5.2 \)[/tex], [tex]\( \text{width} = 4 \)[/tex], [tex]\( \text{height} = 1.5 \)[/tex]
- Length-to-width ratio [tex]\( = \frac{5.2}{4} = 1.3 \)[/tex]
- Width-to-height ratio [tex]\( = \frac{4}{1.5} \approx 2.67 \)[/tex]
- Length-to-height ratio [tex]\( = \frac{5.2}{1.5} \approx 3.47 \)[/tex]
Comparing the ratios with the given prism:
- [tex]\( \frac{5.2}{4} \neq 2.6 \)[/tex]
- [tex]\( \frac{4}{1.5} \neq 0.67 \)[/tex]
- [tex]\( \frac{5.2}{1.5} \neq 1.73 \)[/tex]
Thus, Prism 1 is not similar.
Prism 2:
- Dimensions: [tex]\( \text{length} = 2 \)[/tex], [tex]\( \text{width} = 3 \)[/tex], [tex]\( \text{height} = 6 \)[/tex]
- Length-to-width ratio [tex]\( = \frac{2}{3} \approx 0.67 \)[/tex]
- Width-to-height ratio [tex]\( = \frac{3}{6} = 0.5 \)[/tex]
- Length-to-height ratio [tex]\( = \frac{2}{6} \approx 0.33 \)[/tex]
Comparing the ratios with the given prism:
- [tex]\( \frac{2}{3} \neq 2.6 \)[/tex]
- [tex]\( \frac{3}{6} \neq 0.67 \)[/tex]
- [tex]\( \frac{2}{6} \neq 1.73 \)[/tex]
Thus, Prism 2 is not similar.
Prism 3:
- Dimensions: [tex]\( \text{length} = 1 \)[/tex], [tex]\( \text{width} = 10 \)[/tex], [tex]\( \text{height} = 2.6 \)[/tex]
- Length-to-width ratio [tex]\( = \frac{1}{10} = 0.1 \)[/tex]
- Width-to-height ratio [tex]\( = \frac{10}{2.6} \approx 3.85 \)[/tex]
- Length-to-height ratio [tex]\( = \frac{1}{2.6} \approx 0.38 \)[/tex]
Comparing the ratios with the given prism:
- [tex]\( \frac{1}{10} \neq 2.6 \)[/tex]
- [tex]\( \frac{10}{2.6} \neq 0.67 \)[/tex]
- [tex]\( \frac{1}{2.6} \neq 1.73 \)[/tex]
Thus, Prism 3 is not similar.
After evaluating all the given prisms, none of them exhibit the same set of dimension ratios as the original right rectangular prism. Hence, there is no prism similar to the given right rectangular prism.
The conclusion is:
None of the given prisms are similar to the right rectangular prism with dimensions [tex]\( \text{length} = 5.2 \)[/tex], [tex]\( \text{width} = 2 \)[/tex], and [tex]\( \text{height} = 3 \)[/tex] units.
Therefore, the answer is:
-1
