Question #f65d3

1 Answer
Sep 16, 2017

# [(dh)/dt]_(h=25) = 1/(8pi) ~~ 0.0398 \ cms^-1#

Explanation:

Let us summarise the variables:

# {(r=20, "Radius", (cm)), (h(t), "Height of water at time t", (cm)), (V(t), "Volume of the water at time t", (cm^3)), (t, "time", (s)) :} #

We are given that:

# [(dV)/dt]_(h=25) = 50 \ cm^3s^-1 #

and, we seek the value of:

# [ (dh)/dt ]_(h=25) #

The volume of the water in the cylinder, at time #t#:

# V = pir^2h #
# \ \ \ = pi(20^2)h #
# \ \ \ = 400pi h # ..... [A]

Differentiate [A] Implicitly wrt #t#, and applying the Chain Rule:

# d/dt V = 400pi d/dt h #
# :. (dV)/dt = 400pi (dh)/dt d/(dh) h #
# :. (dV)/dt = 400pi (dh)/dt 1 #

# :. (dV)/dt = 400pi (dh)/dt #

# :. (dh)/dt =1/(400pi ) ( (dV)/dt ) #

So, when #h=25#, we have:

# [(dh)/dt]_(h=25) =50/(400pi ) #