Focus on math practices Reasoning Janine wants to build a model using \( \frac{1}{2} \) inch cubes. How many \( \frac{1}{2} \) inch cubes would she use to build a solid, cube-shaped model with side lengths of 4 inches? Show your work.

To determine how many \( \frac{1}{2} \) inch cubes Janine needs for her cube-shaped model with side lengths of 4 inches, we first need to find the volume of the larger cube. The volume \( V \) of a cube is calculated by using the formula \( V = s^3 \), where \( s \) is the length of a side. Given that the side length \( s \) is 4 inches: \[ V = 4^3 = 64 \text{ cubic inches} \] Next, we calculate the volume of one \( \frac{1}{2} \) inch cube. The volume of the smaller cube is: \[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \text{ cubic inches} \] Now, to find out how many \( \frac{1}{2} \) inch cubes fit into the larger cube, we divide the volume of the larger cube by the volume of the smaller cube: \[ \text{Number of smaller cubes} = \frac{64 \text{ cubic inches}}{\frac{1}{8} \text{ cubic inches}} = 64 \times 8 = 512 \] So, Janine will need 512 \( \frac{1}{2} \) inch cubes to build her cube-shaped model! Building models like this sparks creativity in mathematics, and Janine is on her way to constructing an impressive project!