Answer: a) 2421.743 in³/ min , b) 54437.52 in³/ min
Explanation: The formulae for the volume of a sphere is given below as
V = (4π/3) ×r³ where V =volume of sphere and r = radius of sphere.
By taking the time derivative of the formulae, we have the rate of change of volume with time (we do so by using implicit diffrenciation) , hence we have that
dV/dt = (4π/3) × 3r²×dr/dt
Where dV/dt = rate of change of volume and dr/dt= rate of change of radius.
A)
r = 8 inches, dr/dt = 3 in/min
By substituting this into the formulae, we have that
dV/dt = (4π/3) × 3r²×dr/dt
dV/dt = (4π/3) × 3(8)² × 3
dV/dt = (4π/3) × 576
dV/dt = 2421.743 in³/ min
B)
r = 38 inches, dr/dt = 3 in/min
By substituting this into the formulae, we have that
dV/dt = (4π/3) × 3r²×dr/dt
dV/dt = (4π/3) × 3(38)² × 3
dV/dt = (4π/3) × 12996
dV/dt = 54437.52 in³/ min
