The sphere will have the minimum surface area when the volume is fixed at 1,440 cm³. A sphere has the smallest surface area-to-volume ratio compared to a cube or a cylinder with height equal to diameter.
Step-by-step explanation:
Out of the given shapes, the sphere will have the minimum surface area when the volume is fixed at 1,440 cm³. This is because a sphere has the smallest surface area-to-volume ratio compared to a cube or a cylinder with height equal to diameter.
The formula for the surface area of a sphere is given by 4πr², where r is the radius. The formula for the volume of a sphere is 4/3πr³. Notice that the surface area formula has a square term (r²) while the volume formula has a cubic term (r³). This means that as the radius increases, the surface area increases at a slower rate than the volume, resulting in a smaller surface area-to-volume ratio.
In contrast, a cube and a cylinder with height equal to diameter have larger surface area-to-volume ratios compared to a sphere with the same volume.