1. Part (a)
dV/dt = 2000
“Find dh/dt when r=100, h=0.5, and dr/dt=2.5” u v
(We need the product rule here...)
V = pr2 h
dV/dt = 2pr(dr/dt)h + pr2(dh/dt)
2000 = 2p(100)(2.5)(0.5) + p(100)2(dh/dt)
2000 = 250p + 10,000p (dh/dt)
dh/dt = or cm/min

2. Once the recovery device is in place, dV/dt = 2000 – 400 t
Part (b)
We know that V is a maximum at t=25 min because V”(t) =-200t-1/2, which is always negative. Therefore, V”(25) < 0, so it’s a maximum.
Once the recovery device is in place, dV/dt = 2000 – t
The maximum volume would occur when dV/dt = 0.
400 t = 2000
t = 5
t = 25 minutes

3. The volume of oil at t=25 would be:
Part (c)
The volume of oil at t=25 would be:
400 t
( ) dt
Volume = 60,000 + 25