An inverted conical water tank has a radius of 10m at the top and 24m high. If water flows into the tank at a rate of 20m^3/min, how fast is...

We'll recall the formula that gives the volume of a right
circular cone:

V = pi*h*r^2/3, where h is the height of the
cone and r is the radius of the base.

To determine the
depth of increasing water, at the given rate, we'll have to find out
dh/dt.

First, we'll determine r, using similar
triangles:

r/h = 10/24

r =
5h/12

V =
pi*h*25h^2/3*144

We'll differentiate both sides with
respect to t:

dV/dt =
3*25*pi*h^2/3*144

dV/dt =
(25pi*h^2/144)*(dh/dt)

Since dV/dt = 20m^3/min, when h =
16, we'll get:

20 =
(25pi*16^2/144)*(dh/dt)

(dh/dt) =
144*20/25*256*pi

(dh/dt) =
144/5*64*pi

(dh/dt) =
36/5*16*pi

(dh/dt) =
9/5*4*pi

(dh/dt) =
0.1433

The depth of water is increasing at a
rate of about 0.1433 m/min.