Question

A rectangular prism has a volume of 448 cubic units. How many unit cubes would fit inside?

a) 7
b) 14
c) 28
d) 56

Answer 1

The volume of 448 cubic units can be formed by a rectangular prism with dimensions 14, 8, and 2. Multiplying these dimensions gives 224 unit cubes.Thus the correct option is A.

Step-by-step explanation:

To find the number of unit cubes that can fit inside a rectangular prism, we can use the formula for volume, which is length × width × height. In this case, the volume is given as 448 cubic units. Since a rectangular prism has three dimensions, and we need to find the number of unit cubes, we can express this as:

`\[ \text{Volume} = \text{length} * \text{width} * \text{height} \]`

`\[ 448 = \text{length} * \text{width} * \text{height} \]`

Now, we need to factorize 448 into three positive integers. The prime factorization of 448 is
`(2^6 * 7\)`). To find the dimensions, we can distribute these factors among the three dimensions.

One possible combination is
`\(2^3 * 7\)` for the length and
`\(2^3\)` for both the width and height. Therefore, the dimensions of the rectangular prism are 14, 8, and 2.

Now, to find the number of unit cubes, we multiply these dimensions:

`\[ \text{Number of unit cubes} = \text{length} * \text{width} * \text{height} \]`

`\[ \text{Number of unit cubes} = 14 * 8 * 2 = 224 \]`

Therefore, the correct answer is 224 unit cubes.

Answer 2

56 unit cubes would fit inside (option d)

Step-by-step explanation:

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Given that the volume of the rectangular prism is 448 cubic units, it can be expressed as length × width × height = 448. To determine how many unit cubes would fit inside, this volume represents the total number of unit cubes that can fill the space of the rectangular prism.

Factors of 448 are considered to find the number of unit cubes that can fit, and 448 can be factored as 2 × 2 × 2 × 2 × 7 × 2 = 2^5 × 7 = 32 × 7 = 56. Therefore, 56 unit cubes would fit inside the rectangular prism.(option d)

The volume of a shape represents the total space it occupies, and in the case of a rectangular prism, this volume can be visualized as the number of unit cubes that fill the space within the prism.

Understanding how to calculate volume and relate it to the number of unit cubes required to fill a shape aids in visualizing and conceptualizing spatial measurements, a crucial skill in geometry and practical applications involving measurements and space.