A right cubic prism has edges of 3 1/2 inches. How many cubes with side lengths of 1/2 inches would be needed to fill the prism?

Steps to Solve

1. Calculate the volume of the right cubic prism

The prism is cubic, meaning all sides are the same length. The side length is $3\frac{1}{2}$ inches, which we can write as the improper fraction $\frac{7}{2}$ or the decimal $3.5$. The volume $V_p$ of the prism is the side length cubed: $$V_p = \left(3\frac{1}{2}\right)^3 = \left(\frac{7}{2}\right)^3 = (3.5)^3$$ $$V_p = 3.5 \cdot 3.5 \cdot 3.5 = 42.875 \text{ cubic inches}$$

2. Calculate the volume of one small cube

Each small cube has a side length of $\frac{1}{2}$ inch or 0.5 inches. The volume $V_c$ of one cube is the side length cubed: $$V_c = \left(\frac{1}{2}\right)^3 = (0.5)^3$$ $$V_c = 0.5 \cdot 0.5 \cdot 0.5 = 0.125 \text{ cubic inches}$$

3. Determine 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 one small cube: $$\text{Number of cubes} = \frac{V_p}{V_c} = \frac{42.875}{0.125} = 343$$