Water is leaking out of an inverted conical tank at a rate of 8,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (Round your answer to the nearest integer.)

845758

Step-by-step explanation:

Let  V  be the volume of water in the tank, in  c m 3 ; let  h  be the depth/height of the water, in cm; and let  r  be the radius of the surface of the water (on top), in cm. Since the tank is an inverted cone, so is the mass of water. Since the tank has a height of 6 m and a radius at the top of 2 m, similar triangles implies that  h/ r = 6 /2 = 3

so that  h = 3 r

The volume of the inverted cone of water is then

  =

Now differentiate both sides with respect to time  t (in minutes) to get

   (the Chain Rule is used in this step).

If   is the volume of water that has been pumped in, then

(when the height/depth of water is 2 meters, the radius of the water is

Therefore ≈ 845758 .