Expert: Paul Klarreich - 11/23/2008

Question
A container has the shape of an open right circular cone. The height of the container is 10cm and the diameter or the opening is 10cm. Water in the container is evaporating so that its depth, h, is changing at a constant rate of -3/10 cm/hr.

a)Find the volume V of water in the container when h=5 cm.
b)Find the rate of change of the volume of water in the container, with respect to time, when h=5 cm.

I need help on where to start.  I am usually pretty good at this stuff but I am lost on this one.

Answer
Questioner:   Ray
Category:  Calculus
Private:  No
 
Subject:  Related Rates
Question:  A container has the shape of an open right circular cone. The height of the container is 10cm and the diameter or the opening is 10cm. Water in the container is evaporating so that its depth, h, is changing at a constant rate of -3/10 cm/hr.

>> That is illogical.  Just ask your Physics teacher.

a)Find the volume V of water in the container when h=5 cm.
b)Find the rate of change of the volume of water in the container, with respect to time, when h=5 cm.

I need help on where to start.  I am usually pretty good at this stuff but I am lost on this one.
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Hi, Ray,

Is this your first attempt at R-R problems?  If so, the scheme is something like this:

1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, use your general knowledge and brilliance, do whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.

AND, Please check the archives for other Related Rates examples.  There are a TON of them.  Click BROWSE PAST ANSWERS and look for the subject line Related Rates.
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Start with this:

Since  H (not the same as h) = 10 and  R (not r) = 5, you will always have  h = 2r.

And  V = pi r^2 h

Now you should be okay.

AND I MEAN IT -- READ ALL THE OTHER R-R PROBLEMS ON THIS SITE.