Final answer:
To find the volume of the larger solid with a scale factor of 3:4 when the smaller solid's volume is 54 cubic feet, you cube the scale factor and multiply it by the smaller solid's volume, resulting in 128 cubic feet for the larger solid.
Step-by-step explanation:
When two similar geometric solids have a scale factor of 3:4, it means that the dimensions of the larger solid are each 4/3 times the dimensions of the smaller solid. Since volume increases with the cube of the scale factor, you would cube the scale factor to determine how much larger the volume of the larger solid is compared to the smaller solid. The math for this involves raising the scale factor to the third power: (4/3)^3 = 64/27. To find the volume of the larger solid, multiply the volume of the smaller solid by this ratio:
V_large = V_small × (4/3)^3
V_large = 54 ft³ × (64/27)
V_large = 54 ft³ × (64/27) = 128 ft³
Therefore, the volume of the larger solid is 128 cubic feet, rounded to the nearest tenth as needed.