Final answer:
To determine the height of a rectangular solid with a volume less than a cube with an 8 dm edge, you calculate the cube's volume (512 dm^3) and divide that by the product of the solid's length and width, resulting in a height of less than 8.53 dm.
Step-by-step explanation:
To find the height of the rectangular solid if its volume is smaller than the volume of a cube with an edge length of 8 dm, we first calculate the volume of the cube. The volume of a cube is found by cubing the length of one of its sides (V = s^3), so a cube with a side length of 8 dm has a volume of 512 dm^3 (8×8×8). Next, we need to assume that the volume of our rectangular solid is slightly less than this value.
The volume of a rectangular solid is calculated by the formula V = l×w×h, where l is the length, w is the width, and h is the height. With the base length being 12 dm and the width 5 dm, and knowing the volume is less than 512 dm^3, we can express the height as h < 512 / (12×5). By doing this calculation, we obtain h < 8.53 dm. Therefore, the height of the rectangular solid must be less than 8.53 dm to ensure its volume is smaller than that of the 8 dm cube.