Expert: Ahmed Salami - 1/8/2010

Question
Draining a tank
Water drains from the conical tank shown in the accompanying figure at the rate of 5ft cube/min
a. What is the relation between the variables h and r in the figure?
b. How fast is the water level dropping when h = 6ft?

Please guide me and lead me to the correct answer, i cant solve this question. Thanks.

Answer
Hi Teoh,
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
The smaller cone containing all the water at this point is similar to the larger cone and so the ratio of the dimensions is constant i.e
r/h = R/H
  = 4/10
  = 2/5
r = 2h/5
Now, writing v in terms of h only
v = π(2h/5)².h/3
= π(4h²/25).h/3
= 4πh³/75
Differentiating with respect to time,
dv/dt = (4π/75)(3h²)dh/dt
    = (4πh²/25)dh/dt
dh/dt = (25/4πh²)dv/dt
but dv/dt = 5 ft³/min
So, at h = 6
dh/dt = (25/4π.6²).5
    = 125/144π ft/min
The water level is dropping at about 0.28 ft/min. Actually, both dv/dt and dh/dt are negative since v and h are decreasing with time.

By the way, you havent provided the figure but i think i know exactly what it looks like and the dimensions. If i'm mistaken on R and H being 4 inches and 10 inches respectively, you can get back to me.

Regards