Question

Calculate and compare the volume of the two figures in the image. Choose which statement is true. (1 point) Responses The two figures are not similar because there is no scale factor stated for the volume. The two figures are not similar because there is no scale factor stated for the volume. The two figures are similar because the volumes have a ratio of 5 : 1 . The two figures are similar because the volumes have a ratio of ratio of 5 to 1 . The two figures are not similar because the volumes are not the same. The two figures are not similar because the volumes are not the same. The two figures are similar because the volumes have a ratio of 125 : 1 .

Answer 1

To determine which statement is true regarding the two figures and their volumes, let’s review what is necessary to establish if figures are similar based on their volumes.

1. Understanding Similar Figures: Two three-dimensional figures are similar if they have the same shape but different sizes. This similarity can be quantified through a scale factor. If the scale factor between the two figures is known, the ratio of their volumes will be the cube of the scale factor.

2. Ratio of Volumes: If the volume of the smaller figure is \( V_1 \) and the volume of the larger one is \( V_2 \), the ratio of their volumes can be expressed as: \[ \text{Volume Ratio} = \frac{V_2}{V_1} \]

3. Scale Factor from Volume Ratio: If the volume ratio of two similar figures is \( k \) : 1, the linear dimensions scale factor \( s \) can be found from \( s^3 = k \), where \( k = \frac{V_2}{V_1} \).

4. Evaluating Statements:

   • If the volumes have a ratio of \( 5 : 1 \), this means one volume is \( 5 \) times the other, resulting in a dimension scale factor of \( \sqrt[3]{5} \).
   • If the volumes have a ratio of \( 125 : 1 \), then one volume is \( 125 \) times the other, resulting in a dimension scale factor of \( 5 \).
   • Statements about similarity can also depend on whether the volumes are equal or not, as we can derive from the options presented.

Determining the True Statement:

• If the volumes are equal, the figures can be considered the same (same volume, hence similar).
• If the statement regarding them being similar due to a volume ratio of \( 5 : 1 \) or \( 125 : 1 \) is correct, it indicates that the figures are similar based on the volume scale.

Conclusion: Without the actual volumes presented, you can analyze the statements based on volume ratios, and typically, if the ratio of volumes is a perfect cube, it confirms similarity. If you confirm that the ratio is indeed \( 125 : 1 \), then the statement "The two figures are similar because the volumes have a ratio of 125 : 1" is true.

Therefore, the answer would be: The two figures are similar because the volumes have a ratio of 125 : 1.