Final answer:
The rate of change of the volume of a cube with respect to its side length when the side length is 13 cm is 507 cubic centimeters per centimeter, calculated by the derivative of the volume formula (dV/ds = 3s²).
Step-by-step explanation:
The question asks for the rate of change of the volume with respect to the side length, s, of a cube, when s equals 13 centimeters. To find the rate of change, we need to find the derivative of the volume with respect to s, known as dV/ds.
Given the volume of a cube V = s³, the derivative with respect to s is dV/ds = 3s². When s = 13 cm, the rate of change of the volume is dV/ds = 3*(13²) = 3*169 = 507. So, the rate of change of the volume with respect to side length when s = 13 cm is 507 cubic centimeters per centimeter.