Question

calculate and compare the volume of the two figures H27.5 B27.5 L55 and H5.5 B5.5 L11 in the image. choose which statement is true.
the two figures are similar because the volumes have a ratio of 125 : 1
the two figures are not similar because there is no scale factor stated for the volume
the two figures are not similar because the volumes are not the same
the two fiures are similar because the volumes have a ratio of 5 : 1

Answer 1

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."**