To determine the relationship between the two figures based on their dimensions and volume, we first need to calculate the volume of each figure using the formula for volume of a rectangular prism (Volume = Height × Breadth × Length).
For the first figure (H27.5, B27.5, L55):
\[ \text{Volume}_1 = H_1 \times B_1 \times L_1 = 27.5 \times 27.5 \times 55 \]
\[ \text{Volume}_1 = 27.5 \times 27.5 = 756.25 \] \[ \text{Volume}_1 = 756.25 \times 55 = 41,593.75 \text{ cubic units} \]
For the second figure (H5.5, B5.5, L11):
\[ \text{Volume}_2 = H_2 \times B_2 \times L_2 = 5.5 \times 5.5 \times 11 \]
\[ \text{Volume}_2 = 5.5 \times 5.5 = 30.25 \] \[ \text{Volume}_2 = 30.25 \times 11 = 332.75 \text{ cubic units} \]
Comparing the volumes:
Now we can find the ratio of the volumes of the two figures. \[ \text{Volume ratio} = \frac{\text{Volume}_1}{\text{Volume}_2} = \frac{41,593.75}{332.75} \] Calculating this ratio: \[ \frac{41,593.75}{332.75} \approx 125 : 1 \]
Conclusion:
Now let's examine the statements:
- "the two figures are similar because the volumes have a ratio of 125 : 1" - True, because our calculations show the volume ratio is indeed 125:1.
- "the two figures are not similar because there is no scale factor stated for the volume" - False. The scale factor is implied by their dimensional ratios and the volume ratio.
- "the two figures are not similar because the volumes are not the same" - False. They are still similar even though their volumes are different.
- "the two figures are similar because the volumes have a ratio of 5 : 1" - False. The correct ratio is 125:1.
Thus, the correct statement is: "the two figures are similar because the volumes have a ratio of 125 : 1."
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