Depth of water when the water level is falling the least rapidly
Can someone help me with this problem? Thank you.
A spherical tank of radius 10ft is being filled with water. When it is completely full, a plug at its bottom is removed. According to Torricelli's law, the water drains in such a way that dV/dt = -k*sqrt(y), where V is the volume of water in the tank and k is a positive empirical constant.
a) Find dy/dt as a function of the depth y
b) Find the depth of water when the water level is falling at least rapidly (you will need to compute the derivative of dy/dt with respect to y)
Regarding a), I have done this:
V=1/3*pi*y^2(3a-y) where a is the radius
dV/dt=10*pi*2y(dy/dt) - (1/3)*pi*3y^2(dy/dt)
-k*sqrt(y) = (dy/dt)(20*pi*y - pi*y^2)
And this is what I came up for dy/dt as a function of the depth y
But I am not sure how to proceed with point b) Any help will be apprecited.