Final answer:
The volume of similar figures is proportional to the cube of the scale factor. Using the given surface areas and volume of the larger figure, the volume of the smaller figure is approximately 1272in3.
Step-by-step explanation:
To find the volume of the smaller figure, we can use the concept of similarity. Since the figures are similar, their ratios of corresponding sides will be equal.
We can use the formula for the ratio of volumes of similar figures, which states that the ratio of volumes is equal to the ratio of the cubes of their corresponding sides. Let's denote the ratio of the sides of the smaller figure to the larger figure as 'k'.
SA1 / SA2 = (k2)/(k2) = 486/864 = 9/16
V1 / V2 = (k3)/(k3) = V1 / 1792
Since the ratio of their surface areas is 9/16, we can set up the equation: 9/16 = 486/V2, and solve for V2 to find the volume of the larger figure.
Then we can use the same ratio to find the volume of the smaller figure: 9/16 = V1/1792. Solving for V1, we get V1 = (9/16) * 1792, which is approximately 1008 in³.