Question

The dimensions of a rectangular prism are shown below: Length: 2 and 1 over 4 feet Width: 1 foot Height: 1 and 1 over 4 feet The lengths of the sides of a small cube are 1 over 4 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)

Answer 1

• L=2-1/4=9/4ft
• W=1ft
• H=1-1/4=5/4ft.

Volume of prism

• LWH
• 9/4(5/4)(1)
• 45/16ft³

Volume of cube

• side³
• (1/4)³
• 1/64ft³

Total cubes

• 45/16÷1/64
• 45/16×64
• 45(4)
• 180cubes

#B

Volume of prism

• Volume of one cube×Total cubes
• 180(1/64)
• 45/16ft³

Answer 2

Part A

To find the number of cubes that can be packed into the rectangular prism, calculate the volume of the rectangular prism and the volume of the cube, then divide the volume of the prism by the volume of the cube.

`\begin{aligned}\textsf{Volume of a rectangular prism} & = \sf width * length * height\\\\\implies \sf Volume & = 1 * 2 (1)/(4) * 1 (1)/(4)\\\\& = 1 * (2 * 4+1)/(4) * (1 * 4+1)/(4)\\\\& = 1 * (9)/(4) * (5)/(4)\\\\& = (45)/(16)\end{aligned}`

`\begin{aligned}\textsf{Volume of a cube} & = \textsf{side length}^3\\\\\implies \sf Volume & = \left((1)/(4)\right)^3\\\\& = (1^3)/(4^3)\\\\& = (1 * 1 * 1)/(4 * 4 * 4)\\\\& = (1)/(64)\end{aligned}`

`\begin{aligned}\textsf{Number of cubes} & = \sf \frac{\textsf{Volume of rectangular prism}}{\textsf{Volume of cube}}\\\\& = (45)/(16) / (1)/(64)\\\\& = (45)/(16) * (64)/(1)\\\\& = (45 * 64)/(16 * 1)\\\\& = (2880)/(16)\\\\& = 180\end{aligned}`

Part B

Convert the sides lengths of the rectangular prism into improper fractions with 4 as the denominator:

`\implies \textsf{Length}=2 (1)/(4)=(2 * 4+1)/(4)=(9)/(4)`

`\implies \textsf{Width}=1=(4)/(4)`

`\implies \textsf{Height}=1 (1)/(4)=(1 * 4+1)/(4)=(5)/(4)`

As the side lengths of the cube are 1/4 then the volume of the prism in terms of the small cube is:

⇒ 9 x 4 x 5 = 180 units³

Using the found volume of the prism from part A, the volume of the rectangular prism in terms of a unit cube is:

`\implies \sf (45)/(16)=2.8125\:\:units^3`