Final answer:
The volume of a shampoo bottle can change when squeezed even if its surface area does not change. The volume is related to the shape of the bottle and fluid mechanics principles, not just the surface area.
Step-by-step explanation:
A student asked if the volume of a shampoo bottle remains the same when it is squeezed just because its surface area does not change. The answer is no; even though the surface area might remain constant, the volume can change. For example, if the bottle is made more elongated (like a 'filament'), the surface area can stay the same while the volume decreases. Similarly, changing the shape to a 'sheet' (flattened), again the surface area can remain the same, but the volume can greatly decrease.
This can be understood by considering the formula for the volume of a cylinder, which is the cross-sectional area A times the height h, expressed as V = Ah. If the cylinder is squeezed and the height decreases without the cross-sectional area increasing accordingly, the volume will decrease. This principle is also related to fluid mechanics, where the pressure in a fluid only depends on the depth and not on the shape of the container, and if the shape changes the pressure distribution can change while keeping the same volume.
Therefore, the pressure inside the fluid and the change in shape of the bottle can lead to a change in volume even if the surface area remains the same.