Final answer:
The instantaneous rate of change in the volume of a spherical balloon with a radius increasing at 1 inch per second is found through differentiation to be 256π cubic inches per second when the radius is 8 inches.
Step-by-step explanation:
The student's question involves finding the instantaneous rate of change in volume of a spherical balloon as its radius increases. The volume V of a sphere is given by the formula V = 4/3πr^3, where r is the radius.
To find the rate of change in volume with respect to time, we use differentiation with respect to time (t) on both sides of the volume formula and apply the chain rule.
If the radius r is increasing at a rate of dr/dt = 1 inch/second, then the rate of change in volume dV/dt when the radius is r = 8 inches is calculated as follows:
dV/dt = d/dt (4/3πr^3) = 4πr^2(dr/dt)
Substituting r = 8 inches and dr/dt = 1 inch/second, we get:
dV/dt = 4π(8^2)(1) = 4π(64) = 256π cubic inches per second
Therefore, the instantaneous rate of change in volume for the balloon is 256π cubic inches per second when its radius is 8 inches.