Expert: Paul Klarreich - 11/22/2008

Question
An oil storage tank has the shape of a paraboloid with a radius of 5ft and a height of 9ft. Oil flows into the tank at the rate of 8 cubic feet/min.  The volume of the tank is V=(50*pi*h^(3/2))/3.

a)Find the volume of a full tank
b)How long would it take to fill the tank if it was initially empty?
c)How fast is the height of oil increasing when h=4?

I am completely lost on what to do.  I am pretty good at this stuff but I don't even know where to start on this one.

Answer
Questioner:   smiff
Category:  Calculus
Private:  No
 
Subject:  Related Rates involving paraboloid
Question:  An oil storage tank has the shape of a paraboloid with a radius of 5ft and a height of 9ft. Oil flows into the tank at the rate of 8 cubic feet/min.  The volume of the tank is V=(50*pi*h^(3/2))/3.

a)Find the volume of a full tank
b)How long would it take to fill the tank if it was initially empty?
c)How fast is the height of oil increasing when h=4?

I am completely lost on what to do.  I am pretty good at this stuff but I don't even know where to start on this one.
...............................................
Hi, Smiff,

Usually you get started by setting up coordinates and deducing some equation. Suppose we assume that the parabola (yes, parabola -- we'll worry about the paraboLOID later.)  has its vertex at (0,0).

Then its equation is   y = ax^2, and all we need is one point.

Such as  (5,9)??

Ok, then   9 = a(25), so  a = 9/25.

y = 9x^2/25

Now the area of a cross-section is  pi x^2.  There probably is a formula for this, but we can figure it out ourselves.  All we have to do is integrate:
................
a) Using the disk method:

sample disk dV = pi x^2 dy,  y = 0 to 9,

So integrate:

{9
|    pi 25y/9 dy
}0

pi 25y^2/18 , from 0 to 9

= pi 25(81)/18 = pi 25(9)/2 = 225pi/2.
................................
Suggestion:  Try doing it with cylindrical shells, just for practice.  (it works -- I tried it.)
...................................
b) I think you can handle that.

...............................
c) Now you need the volume of the water already in the tank.

You have  

dV/dt = 8
dy/dt TO BE FOUND.

You need a formula for V = f(y), which is:

(TADA!!!!!!!!..............)

pi 25y^2/18, remember?

So now  V = 25pi y^2  and you can do the rest of this as any other Related Rates example.