Question

8. For the two designs in question #7, which one holds more volume?
*
the rectangular prism with dimensions of 3 in. x 1 in. x 4 in.
the cube with a side length from the answer in #7

Answer 1

To determine which design holds more volume, let's first calculate the volumes of each design.

1. **Volume of the Rectangular Prism**:
- The dimensions are given: 3 inches by 1 inch by 4 inches.
- Volume (V) is calculated using the formula for the volume of a rectangular prism:
\[ V = \text{length} \times \text{width} \times \text{height} \]
\[ V = 3 \text{ in.} \times 1 \text{ in.} \times 4 \text{ in.} \]
\[ V = 12 \text{ cubic inches} \]

2. **Volume of the Cube**:
- The side length needs to be determined from the solution to question #7. For the purpose of this solution, let’s assume in question #7 it was determined a cube that holds the same total volume as 12 cubic inches (since it's for comparison).

If the cube is to match the volume of 12 cubic inches, we set up the formula for the volume of a cube (where \( s \) is the side length):
\[ V = s^3 \]
\[ s^3 = 12 \text{ cubic inches} \]

Solving for \( s \):
\[ s = \sqrt[3]{12} \]

The approximate value is:
\[ s \approx 2.29 \text{ inches} \]

- To confirm, we calculate the volume again using this side length:
\[ V = (2.29 \text{ inches})^3 \approx 12 \text{ cubic inches} \]

Since the side length of the cube derived from a volume of 12 cubic inches matches the volume of the rectangular prism directly, both designs (rectangular prism and the cube) have equal volume. Hence, neither design holds more volume than the other; both have approximately 12 cubic inches of volume.