The dimensions of a rectangular prism are shown below:
Length: 1 and 1 over 3 feet
Width: 1 foot
Height: 2 and 1 over 3feet
The lengths of the sides of a small cube are 1 over 3 foot each.
Part A: How many small cubes can be packed in the rectangular prism? Show your work. (5 points)
Part B: Use the answer obtained in part A to find the volume of the rectangular prism in terms of the small cube and a unit cube. (5 points)
Part A:
To find out how many small cubes can be packed in the rectangular prism, we need to determine the number of small cubes that can fit along each side.
1. First, let's find out the number of small cubes that can fit along the length of the rectangular prism. Since the length of the prism is 1 and 1/3 feet and each small cube has a side length of 1/3 foot, we can divide the length of the prism by the side length of the small cube: (1 and 1/3) ÷ (1/3).
(4/3) ÷ (1/3) = 4.
Therefore, we can pack 4 small cubes along the length of the prism.
2. Next, let's determine the number of small cubes that can fit along the width of the rectangular prism. The width of the prism is 1 foot, and each small cube also has a side length of 1/3 foot. So we divide the width of the prism by the side length of the small cube: 1 ÷ (1/3).
1 ÷ (1/3) = 3.
Hence, we can fit 3 small cubes along the width of the prism.
3. Finally, let's calculate the number of small cubes that can fit along the height of the rectangular prism. The height of the prism is 2 and 1/3 feet, and each small cube has a side length of 1/3 foot. Thus, we divide the height of the prism by the side length of the small cube: (2 and 1/3) ÷ (1/3).
(7/3) ÷ (1/3) = 7.
Therefore, we can pack 7 small cubes along the height of the prism.
Now, to find the total number of small cubes that can be packed in the rectangular prism, we multiply the number of cubes along each side:
4 (length) × 3 (width) × 7 (height) = 84.
So, 84 small cubes can be packed in the rectangular prism.
Part B:
To find the volume of the rectangular prism in terms of the small cube and a unit cube, we need to determine the number of unit cubes that make up the rectangular prism.
Since we know that each small cube is equal to 1/3 of a unit cube, we can find the volume of the rectangular prism in terms of unit cubes by dividing the total number of small cubes (found in Part A) by 1/3:
84 ÷ (1/3) = 84 × 3.
84 × 3 = 252.
Thus, the volume of the rectangular prism in terms of the small cube and a unit cube is 252 unit cubes.