Question

At time t=0 the bottom plug(at the vertex) of a full conical water tank 16ft high is removed. After 1 hour the water in the tank is 9ft deep. When will the tank be empty?

Answer 1

Assume that water is an incompressible fluid.
Now the rates of change of each quantity like radius, height and volume will be constant.
Consider the rate of change of height of water w.r.t time.
dh/dt = constant
Now, if you see 7ft of water has decreased in 1hour according to the question.
Hence rate of decrease of height of water is 7ft/hr.
Hence, dh/dt = -7 (minus, since it is decreasing)
So dh = -7dt
Now integrate.
Limits of h will be 16ft to 0ft (till the container empties)
Limits of t will be 0 to t(hours)
Answer is 2.285 hours