Question

oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius r and height changing h is given by V= pie times r to the second power times h)

a) at the instant when the radius of oil slick is 100 cenimetters and the hieght is 0.5 cm the radius is increasing at the rate of 2.5 cm per min. at this instant what is the rate of change of the height of oil slick with respect to time in cm per min

Answer 1

let the thickness of the oil slick be h cm
let the radius be r cm
let the volume be V cm^3

given:
dV/dt = 2000cm^3/min
when r=100 cm, dr/dt = 2.5 cm/min and

V=pi(r^2)h

dV/dt = pi(r^2)dh/dt + 2pi(r)(h)dr/dt
2000 = pi)100^2)dh/dt + 2pi(100)(.5)(2.5)

solve for dh/dt