Expert: Ahmed Salami - 1/6/2010

Question
Water is draining at the rate of 48 pie ft. cubed from the vertex at the bottom of a conical tank whose diameter at its base is 40 feet and whose height is 60 feet.
a. Find an expression for the volume of water in the tank in terms of its radius at the surface of the water.

Answer
Hi Stephanie,
The volume of a conical tank with radius R and height H is
V = πR²H/3
If water is draining from the tank, the volume of water at any point where the radius at the surface of the water is r and with height h is given by
v = πr²h/3
Now, the imaginary cone containing all the water at this point is similar to the conical tank and so the ratio of the dimensions is constant i.e
h/r = H/R
   = 60/20
   = 3
h = 3r
Therefore,
v = πr²(3r)/3
 = πr³

Regards