The volumes of the resulting prisms will not be equal. The cube's volume will be double the cylinder's volume after these modifications.
**Current scenario:**
- Let the side length of the cube be `s` and the radius of the cylinder be `r`.
- Volume of the cube: `V_cube = s^3`
- Volume of the cylinder: `V_cylinder = πr^2h` (assume cylinder height remains the same)
**Scenario 1: Doubling two sides of the cube:**
- New side length of the cube: `2s`
- New volume of the cube: `V_cube_new = (2s)^3 = 8s^3`
**Scenario 2: Doubling the diameter of the cylinder:**
- New diameter: `2d = 2 * 2r = 4r`
- New radius: `r_new = d/2 = 4r/2 = 2r`
- New volume of the cylinder: `V_cylinder_new = π(2r)^2h = 4πr^2h`
**Comparing new volumes:**
- Ratio of new cube volume to old cube volume: `V_cube_new / V_cube = 8s^3 / s^3 = 8`
- Ratio of new cylinder volume to old cylinder volume: `V_cylinder_new / V_cylinder = 4πr^2h / πr^2h = 4`
Conclusion:
- The volume of the cube increases by a factor of 8 when two sides are doubled.
- The volume of the cylinder increases by a factor of 4 when the diameter is doubled.
Therefore, the volumes of the resulting prisms will not be equal. The cube's volume will be double the cylinder's volume after these modifications.