Final answer:
The rate of change of the volume with respect to the side length is 768 cm³/cm.
Step-by-step explanation:
To find the rate of change of the volume with respect to the side length of a cube, we need to find the derivative of the volume formula V = s³ with respect to s. This derivative represents the rate of change of the volume when the side length changes.
Using the power rule for differentiation, we differentiate V = s³ with respect to s to get dV/ds = 3s². This means that for every increase of 1 cm in the side length, the volume increases by 3s² cm³.
When s = 16 cm, the rate of change of the volume with respect to the side length is dV/ds = 3(16)² = 768 cm³.