How many different rectangular prisms can be made using 36 cubes if the height is 2 cubes?
To determine the number of different rectangular prisms that can be made using 36 cubes where the height is fixed at 2 cubes, we can consider the possible combinations of the remaining 34 cubes for the length and width.
Since a rectangular prism has 6 faces, we need to distribute the 34 remaining cubes among these faces. We can visualize this by considering all possible factors of 34 and pairing them to find different combinations.
The factors of 34 are: 1, 2, 17, and 34.
Possible combinations of pairs of factors are:
(1, 34), (2, 17), (17, 2), (34, 1).
Each pair represents a possible combination of length and width. For example, (1, 34) represents a rectangular prism with a length of 1 cube and a width of 34 cubes.
Therefore, there are 4 different rectangular prisms that can be made using 36 cubes, where the height is fixed at 2 cubes.
To determine the number of different rectangular prisms that can be made using 36 cubes with a height of 2 cubes, we need to find all the possible combinations of length and width.
First, let's list all the positive factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Since the height is fixed at 2 cubes, we have to distribute the remaining 34 cubes between the length and width.
To place the cubes, we can imagine a grid where the length is represented by the columns and the width is represented by the rows. We need to find all the pairs of numbers whose product is equal to 34.
Here are the possible combinations of length and width:
1. Length = 1, Width = 34 (1 * 34 = 34)
2. Length = 2, Width = 17 (2 * 17 = 34)
3. Length = 3, Width = 34/3 (3 * 34/3 = 34)
4. Length = 4, Width = 34/4 (4 * 34/4 = 34)
5. Length = 6, Width = 34/6 (6 * 34/6 = 34)
6. Length = 9, Width = 34/9 (9 * 34/9 = 34)
7. Length = 12, Width = 34/12 (12 * 34/12 = 34)
8. Length = 18, Width = 34/18 (18 * 34/18 = 34)
9. Length = 34, Width = 1 (34 * 1 = 34)
So, there are nine different rectangular prisms that can be made using 36 cubes with a height of 2 cubes.
height is 2, leaving a cross-section of 18
That can be 1x18 or 2x9 or 3x6