To calculate and compare the volumes of the two figures, we need to find the volume of each figure individually. The figures shown appear to be triangular prisms, and the formula for the volume of a prism is:

$V = \text{Base Area} \times \text{Height}$

Step 1: Volume of Figure A

For Figure A, the dimensions are:

• The base is a right triangle with both legs 27.5 ft.
• The height of the prism (perpendicular to the triangular base) is 55 ft.

Calculating the area of the triangular base (Figure A):

Since the base is a right triangle, the area of the triangular base is:

$\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}$

For Figure A:

$\text{Base Area} = \frac{1}{2} \times 27.5 \times 27.5 = \frac{1}{2} \times 756.25 = 378.125 \, \text{ft}^2$

Calculating the volume (Figure A):

$V_A = \text{Base Area} \times \text{Height} = 378.125 \times 55 = 20,796.875 \, \text{ft}^3$

Step 2: Volume of Figure B

For Figure B, the dimensions are:

• The base is a right triangle with both legs 5.5 ft.
• The height of the prism (perpendicular to the triangular base) is 11 ft.

Calculating the area of the triangular base (Figure B):

$\text{Base Area} = \frac{1}{2} \times 5.5 \times 5.5 = \frac{1}{2} \times 30.25 = 15.125 \, \text{ft}^2$

Calculating the volume (Figure B):

$V_B = \text{Base Area} \times \text{Height} = 15.125 \times 11 = 166.375 \, \text{ft}^3$

Step 3: Compare the Volumes

Now we compare the volumes of the two figures:

$\text{Ratio of Volumes} = \frac{V_A}{V_B} = \frac{20,796.875}{166.375} \approx 125$

Conclusion:

The ratio of the volumes is approximately 125:1, meaning that the two figures are similar because the volumes have a ratio of 125 to 1.

Thus, the correct statement is:

• "The two figures are similar because the volumes have a ratio of 125 : 1."

Would you like further details or have any other questions?

Here are 5 related questions you might find interesting:

1. How can you verify the figures are similar based on their dimensions, not just their volumes?
2. What is the scale factor of the linear dimensions of the two figures?
3. How do changes in linear dimensions affect the volume of geometric figures?
4. What would happen to the volume ratio if the height of one figure changed?
5. Can the same volume ratio be used to compare other shapes like cylinders?

Tip: When comparing similar figures, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions.