Figure a 27.5ft 27.5ft 55ft

Question

Figure b 5.5ft 5.5ft 11ft

Calculate and compare the volume of the two figures in the image. Choose which statement is true.
(1 point)
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 figures are similar
• because the volumes have a ratio of 5 : 1.
The two figures are similar
• because the volumes have a ratio of 125 : 1.

Answer 1

To determine if the two figures are similar and to find the volume ratio, let's calculate the volumes of both figures.

Volume Calculation

1. Figure A: Dimensions = 27.5ft x 27.5ft x 55ft

   • Volume of Figure A = Length × Width × Height \[ V_A = 27.5 \times 27.5 \times 55 \]
   • Calculating it: \[ V_A = 27.5^2 \times 55 = 756.25 \times 55 = 41593.75 \text{ cubic feet} \]

2. Figure B: Dimensions = 5.5ft x 5.5ft x 11ft

   • Volume of Figure B = Length × Width × Height \[ V_B = 5.5 \times 5.5 \times 11 \]
   • Calculating it: \[ V_B = 5.5^2 \times 11 = 30.25 \times 11 = 332.75 \text{ cubic feet} \]

Ratio of Volumes

Next, we will find the ratio of the volumes \(V_A\) to \(V_B\): \[ \text{Ratio} = \frac{V_A}{V_B} = \frac{41593.75}{332.75} \approx 125:1 \]

Determine Similarity

To check the similarity:

• The ratio of the lengths of corresponding sides between the figures is: \[ \text{Side Ratio} = \frac{27.5}{5.5} = 5 \] Thus, because the lengths share the same ratios, the figures are similar in shape.

Conclusion

The correct statement based on the calculations is: The two figures are similar because the volumes have a ratio of 125 : 1.