Final answer:
To find the rate of change of the radius, we can use the formula for the volume of a sphere and differentiate it with respect to time. Substituting the given values into the equation allows us to solve for the rate of change of the radius.
Step-by-step explanation:
To find the rate of change of the radius, we can use the formula for the volume of a sphere, which is V = (4/3)πr³. By differentiating this formula with respect to time, we can find the rate of change of the volume with respect to time, dV/dt. Since the volume is increasing at a rate of 6 in³/min, we have dV/dt = 6 in³/min.
Next, we can differentiate the volume formula with respect to the radius, giving us dV/dr = 4πr². Using this information, we can solve for dr/dt in the equation dV/dt = (dV/dr)(dr/dt). Since we know dV/dt and we want to find dr/dt when r = 3 in, we can substitute these values into the equation to solve for dr/dt.
Substituting dV/dt = 6 in³/min and r = 3 in into the equation, we have 6 in³/min = (4π(3 in)²)(dr/dt). Solving for dr/dt, we get dr/dt = (6 in³/min) / (4π(3 in)²). Evaluating this expression gives us the rate of change of the radius when the radius is 3 in.