Aroblem 7 a. Calculate the number of cubes with edge length \( \frac{1}{2} \mathrm{~cm}^{2} \) that fit in this prism. b. What is the volume of the prism in \( \mathrm{cm}^{3} \) ? Show your reasoning if you are stuck, think about how many cubes with \( \frac{1}{2}-\mathrm{cm}^{3} \) edge lengths fit into \( 1 \mathrm{~cm}^{3} \).

To solve the problem, we need to break it down into two parts: calculating the number of cubes that fit in the prism and finding the volume of the prism. ### Part a: Calculate the number of cubes with edge length \( \frac{1}{2} \, \text{cm} \) that fit in the prism. 1. **Volume of a single cube**: The volume \( V \) of a cube is given by the formula: \[ V = \text{edge length}^3 \] For a cube with edge length \( \frac{1}{2} \, \text{cm} \): \[ V = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \, \text{cm}^3 \] 2. **Volume of the prism**: To find the number of cubes that fit into the prism, we first need to know the volume of the prism. Let's denote the volume of the prism as \( V_{\text{prism}} \). 3. **Number of cubes that fit in the prism**: The number of cubes \( N \) that can fit into the prism is given by: \[ N = \frac{V_{\text{prism}}}{V_{\text{cube}}} = \frac{V_{\text{prism}}}{\frac{1}{8}} = 8 \times V_{\text{prism}} \] ### Part b: What is the volume of the prism in \( \text{cm}^3 \)? To find the volume of the prism, we need its dimensions (length, width, height). However, since the dimensions are not provided in the problem, we can only express the volume in terms of the dimensions. 1. **Volume of the prism**: The volume \( V_{\text{prism}} \) of a prism is calculated as: \[ V_{\text{prism}} = \text{Base Area} \times \text{Height} \] Without specific dimensions, we cannot calculate a numerical value for the volume. 2. **Assuming we have the volume**: If we assume the volume of the prism is \( V_{\text{prism}} \, \text{cm}^3 \), then the number of cubes that fit in the prism would be: \[ N = 8 \times V_{\text{prism}} \] ### Conclusion To summarize: - The volume of a single cube with edge length \( \frac{1}{2} \, \text{cm} \) is \( \frac{1}{8} \, \text{cm}^3 \). - The number of such cubes that fit into the prism is \( 8 \times V_{\text{prism}} \), where \( V_{\text{prism}} \) is the volume of the prism in \( \text{cm}^3 \). If you provide the dimensions of the prism, I can help you calculate the exact volume and the number of cubes that fit inside it.