Expert: Paul Klarreich - 1/30/2010

Question
A flask in shape of a cone of height 20cm and radius 8cm is held vertex downwards.
Show that when the depth of water in flask is x cm, the volume of water is 4/(75 ) πx^3  〖cm〗^3.
Water leaks out from the vertex at rate of 2〖cm〗^3. Find the rate of change of depth of the water when the depth is 10cm.

Sir, could you please help me..I don't really understand what is the meaning of depth. Throughout the cone, even when the height is decreasing, will the radius of cone remain constant sir?

Thank you a lot. Good day.

Answer
Questioner: Abirami
Country: Malaysia
Category: Calculus
Private: No
Subject: Rate of CHange
Question: A flask in shape of a cone of height 20cm and radius 8cm is held vertex downwards.
Show that when the depth of water in flask is x cm, the volume of water is 4/(75 ) πx^3  〖cm〗^3.
Water leaks out from the vertex at rate of 2〖cm〗^3. Find the rate of change of depth of the water when

the depth is 10cm.

Sir, could you please help me..I don't really understand what is the meaning of depth. Throughout the

cone, even when the height is decreasing, will the radius of cone remain constant sir?

Thank you a lot. Good day.
..........................................

In this case you have two 'cones' -- the flask, and the water currently in it.

The conical 'water' is similar (look up similarity in your geometry book) to the flask cone.

So, since the flask has  height = 2.5 * radius, so does the water cone.

Since the water cone height is decreasing , so will the radius.

Now:

1. The 'depth' of the water is the height of the water cone.

2. This becomes a standard Related Rates problem.  Look those up on the site -- there are loads of them.  See, in particular:

http://en.allexperts.com/q/Calculus-2063/2009/11/Related-Rates-87.htm