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 = 0.038 or 0.039 cm/min

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 – 400 t The maximum volume would occur when dV/dt = 0. 400 t = 2000 t = 5 t = 25 minutes

The volume of oil at t=25 would be: Part (c) The volume of oil at t=25 would be: 400 t (2000 - ) dt Volume = 60,000 + 25