Jesse has constructed a huge cylindrical can with a diameter of 60 ft. The can is being filled with water at a rate of 450 ft3/min. How fast is the depth of the water increasing? (Hint: The volume of water in the cylinder is determined by πr2h where r is the radius and h is the depth of the water )

Answer 1

the rate of change of volume is the surface area times the rate of change of height

450 ft^3/min = surface area * dh/dt
surface area = pi r^2

450 = pi (30^2) dh/dt

dh/dt = .159 ft/min

You could do this by saying
V = pi r^2 h
dV/dh = pi r^2
dV/dh*dh/dt = pi r^2 dh/dt
chain rule
dV/dt = pi r^2 dh/dt

but most of us just look at the lake rising an inch and seeing that the added volume is the area times one inch

Answer 2

Thanks Damon, that really clears it up for me