A right cubic prism has edges of 3 1/4 inches. How many cubes with side lengths of 1/4 inches would be needed to fill the prism?

Steps to Solve

1. Calculate the volume of the right cubic prism

The volume of a cube is given by $V = s^3$, where $s$ is the side length. The side length of the prism is $3\frac{1}{4}$ inches, which can be written as an improper fraction: $3\frac{1}{4} = \frac{3 \cdot 4 + 1}{4} = \frac{13}{4}$. Thus, the volume of the prism is: $$V_{prism} = \left(\frac{13}{4}\right)^3 = \frac{13^3}{4^3} = \frac{2197}{64} \text{ cubic inches}$$

2. Calculate the volume of the small cube

The side length of the small cube is $\frac{1}{4}$ inches. So, the volume of the small cube is: $$V_{cube} = \left(\frac{1}{4}\right)^3 = \frac{1^3}{4^3} = \frac{1}{64} \text{ cubic inches}$$

3. Find the number of small cubes needed

To find how many small cubes are needed to fill the prism, divide the volume of the prism by the volume of the cube: $$ \text{Number of cubes} = \frac{V_{prism}}{V_{cube}} = \frac{\frac{2197}{64}}{\frac{1}{64}} = \frac{2197}{64} \cdot \frac{64}{1} = 2197 $$