The right rectangular prism is packed with unit cubes of the appropriate unit fraction edge lengths. Find the volume of the right rectangular prism in centimeters. (Figure not to scale)
1х
x
A
B
С
1
108
3
cm?
1
133 cm?
3
105 cm?
D
100 cm
We can use the information given in the figure to find the dimensions of the prism. The three edges shown as 1, x, and 3 must correspond to the three pairwise dimensions of the prism. Therefore, we have:
- The bottom face has dimensions x by 3.
- The height of the prism is 1.
- The prism contains a total of 108 small unit cubes.
Since the volume of the prism is given by the product of its three dimensions, we need to determine the value of x that satisfies all three conditions above. Since 108 is divisible by 3, we know that the bottom face contains a multiple of 3 unit cubes. Therefore, x must also be a multiple of 3. We can try different values of x until we find one that works:
- If x=3, then the bottom face contains 3x3=9 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 9x1=9, which is too small.
- If x=6, then the bottom face contains 6x3=18 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 18x1=18, which is still too small.
- If x=9, then the bottom face contains 9x3=27 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 27x1=27, which is still too small.
- If x=12, then the bottom face contains 12x3=36 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 36x1=36, which is still too small.
- If x=15, then the bottom face contains 15x3=45 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 45x1=45, which is still too small.
- If x=18, then the bottom face contains 18x3=54 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 54x1=54, which is still too small.
- If x=21, then the bottom face contains 21x3=63 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 63x1=63, which is too large.
- If x=20, then the bottom face contains 20x3=60 unit cubes.
- The height of the prism is 1, so the total number of unit cubes is 60x1=60, which is the correct amount.
Therefore, the dimensions of the prism are 20x3x1, and the volume is:
V = 20 x 3 x 1 = 60 cubic centimeters.