Final answer:
The expression dv/dr represents the instantaneous rate of change of the volume of a sphere with respect to its radius, meaning how much the volume changes for a small change in the radius. The correct answer is option 1: The rate of change of the volume of the sphere.
Step-by-step explanation:
The expression dv/dr represents the instantaneous rate of change of the volume v of a sphere with respect to its radius r. This rate of change can be understood as how much the volume of the sphere will change in response to a very small change in the radius of the sphere. Specified as dv/dr = a v/r, it indicates the proportionality between the differential change in volume dv and the differential change in radius dr. To answer the question, dv/dr represents the rate of change of the volume of the sphere (option 1).
The formula for the volume of a sphere is 4/3 π r^3, which can be differentiated with respect to r to find the rate of change of the volume. The surface area of a sphere is given by 4 π r^2. Therefore, the expression that represents the volume of a sphere is 4/3 π r^3, which helps clarify that dv/dr is concerned with volume, not surface area, radius, or circumference.