Math / Geometry

QuestionHow many cubes with a side length of $1 / 4$ , will fit inside the rectangular prismbelow?

Studdy Solution

STEP 1

1. The rectangular prism has dimensions $\frac{5}{4}$ units, $\frac{5}{4}$ units, and $\frac{1}{4}$ units.
2. Each cube has a side length of $\frac{1}{4}$ unit.
3. The cubes are packed without any gaps or overlaps.

STEP 2

1. Calculate the volume of the rectangular prism.
2. Calculate the volume of one cube.
3. Determine how many cubes fit inside the prism by dividing the volume of the prism by the volume of one cube.

STEP 3

Calculate the volume of the rectangular prism using the formula for volume:
$V_{\text{prism}} = \text{length} \times \text{width} \times \text{height}$ $= \frac{5}{4} \times \frac{5}{4} \times \frac{1}{4}$

STEP 4

Calculate the volume of one cube using the formula for volume:
$V_{\text{cube}} = \left(\frac{1}{4}\right)^3$

STEP 5

Determine how many cubes fit inside the prism by dividing the volume of the prism by the volume of one cube:
$\text{Number of cubes} = \frac{V_{\text{prism}}}{V_{\text{cube}}}$
Calculate the volumes:
$V_{\text{prism}} = \frac{5}{4} \times \frac{5}{4} \times \frac{1}{4} = \frac{25}{64}$ $V_{\text{cube}} = \left(\frac{1}{4}\right)^3 = \frac{1}{64}$
$\text{Number of cubes} = \frac{\frac{25}{64}}{\frac{1}{64}} = 25$
The number of cubes that will fit inside the rectangular prism is:
$\boxed{25}$

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