A roll of wire with a length of 55 cm was used to make the framework for a cube. The volume of the cube is a perfect cube. If the wire is used to the maximum, calculate the length, in cm, remainder of the wire.

To find the length of the wire remaining after making the framework for a cube, we first need to determine the dimensions of the cube that can be constructed using the wire. 1. **Understanding the framework of a cube**: A cube has 12 edges. If we denote the length of each edge of the cube as \( a \), the total length of wire needed to construct the framework of the cube is given by: \[ \text{Total length of wire} = 12a \] 2. **Using the available wire**: We have a roll of wire with a length of 55 cm. Therefore, we set up the equation: \[ 12a \leq 55 \] To find the maximum integer value of \( a \), we can solve for \( a \): \[ a \leq \frac{55}{12} \approx 4.5833 \] Since \( a \) must be a whole number, the maximum integer value for \( a \) is 4 cm. 3. **Calculating the total length of wire used**: If \( a = 4 \) cm, then the total length of wire used is: \[ 12a = 12 \times 4 = 48 \text{ cm} \] 4. **Calculating the remaining length of wire**: The remaining length of wire after constructing the cube is: \[ \text{Remaining wire} = 55 \text{ cm} - 48 \text{ cm} = 7 \text{ cm} \] 5. **Verifying the volume of the cube**: The volume \( V \) of the cube with edge length \( a \) is given by: \[ V = a^3 = 4^3 = 64 \text{ cm}^3 \] Since 64 is a perfect cube, this condition is satisfied. Thus, the length of the wire remaining after making the framework for the cube is \( \boxed{7} \) cm.