To find the rate at which the volume of the snowball is decreasing, we need to use the formula for the volume of a sphere and differentiate it with respect to time.Given:
Rate of change of radius: dr/dt = -0.4 cm/min (negative because the radius is decreasing)
Radius: r = 12 cmThe volume of a sphere is given by the formula:
V = (4/3)πr^3Differentiating both sides of the equation with respect to time (t), we get:dV/dt = 4πr^2 (dr/dt)Substituting the given values:
dV/dt = 4π(12)^2 (-0.4)
= 4π(144) (-0.4)
= -576π cm^3/minThe rate at which the volume of the snowball is decreasing when the radius is 12 cm is approximately -576π cm^3/min. Since the question asks for a positive rate, we take the absolute value of the result:|dV/dt| = 576π cm^3/minTherefore, the volume of the snowball is decreasing at a rate of approximately 576π cm^3/min.