Final answer:
To find the number of rectangular prisms, one must consider all dimensions of smaller prisms that fit within the given 3x4x5 rectangular prism, including the single unit prisms. By calculating the possible lengths, widths, and heights and summing their series, and then multiplying them together, you can obtain the total count.
Step-by-step explanation:
The question is asking to find the total number of rectangular prisms that can be formed within a larger rectangular prism with the dimensions 3 × 4 × 5 using 60 unit cubes. To find this, we need to consider all the different dimensions of smaller prisms that can fit within the larger one. This includes counting each individual unit cube as a rectangular prism itself.
The larger prism has 3 possible lengths (1, 2, 3), 4 possible widths (1, 2, 3, 4), and 5 possible heights (1, 2, 3, 4, 5). To get the count for each dimension, we subtract the number of possible dimensions by 1 and then sum the series from 1 to the resulting number. For length, that would be 1 + 2 + 3; for width, 1 + 2 + 3 + 4; for height, 1 + 2 + 3 + 4 + 5. Multiplying these sums together gives us the total count.
Adding the count of the full prism itself and those of the unit cubes, we get the total number of rectangular prisms as an answer.