differentiation question...?

Question

a cylindrical jar of radius 5cm contains water to a depth of 4cm. the water is then poured at a steady rate into a hemisphere bowl of radius 6cm. after t seconds, the depth of water in the bowl is x cm and the volume, Vcm^3, of water that has been transferred is given by V= [pi x^2(18-x)]/3 cm^3. given that all the water is transferred in 25 seconds, find
(i) dV/dt in terms of pi
(ii) the rate at which x is increasing when x=3

Answer 1

Volume in the jar = pi*25*5==100pi cm^3
as it is transfered at a steady state dV/dt = 4*pi cm^3/s
dx/dt= dx/dV*dV/dt
and dx/dV= 1/(dV/dx)
dV/dx=1/3pi[2x(18-x)-x^2]= 1/3pi(-3x^2+36x)
so dx/dt = 3/pi*(-3x^2+36x) * 4pi = 12/(-3x^2+36x)
At x= 3
dx/dt= 12(-27+108)=0.148cm/s