Related Rates and a Decreasing Radius

Date: 11/18/98 at 11:14:29
From: Chad Bartel
Subject: Calculus Problem

Dr Math:

I'm stumped on a recent calculus problem. Could you help?

A hemispherical lump of nitroglycerin is combusting evenly at a 
rate of 2 cu. cm/sec. How fast is the lump's radius decreasing when 
there is 9pi/4 cu.cm left?

Any insight would be a big help to me and my classmates. Thanks.

Date: 11/18/98 at 12:50:12
From: Doctor Sam
Subject: Re: Calculus Problem

Chad,

This is a typical related rates question. The problem is for you to 
find a relation between the rate at which this volume is changing 
(given as 9pi/4) and the rate at which the radius is changing (which is 
what we need to find.)

"Rate" means "slope" means "derivative." In calculus notation the 
problem reads:

   A hemispherical lump of nitroglycerin is combusting evenly making 
   the volume change at a rate of dV/dt = -2 cu.cm/sec. Find dr/dt 
   when V = 9 pi/4 cu.cm.

The reason that I write dV/dt = -2 is that the volume is decreasing.  
If it were increasing I would write dV/dt = 2.

In order to find derivatives you need something to differentiate! So 
the first step is to find a formula for the volume of a hemisphere.  
Since the volume of a sphere is V = (4/3)pi r^3, the volume of a 
hemisphere is V = (2/3) pi r^3.

This is a formula for V in terms of R. In this problem (and in most 
related rates problems) all of the quantities are varying with time, 
so although you cannot see any t's in the equation both V and r are 
functions of time.  

Since we want to find dr/dt, we differentiate both sides of this 
equation implicitly with time. You will end up with something like 
this:

   dV/dt = (something)dr/dt

The dr/dt comes from differentiating r^3 implicitly.

Now you know the value of dV/dt and you want to find the value of 
dr/dt. Unfortunately, the terms inside the parentheses, the 
"something," includes r, and you are not given the value of r.  

But you are given the value of V. This is a way of fixing the problem 
at a certain instant of time, when the volume is 9 pi/4.

But we now have a formula for the volume of a hemisphere. If you know 
V then you can use this formula to find a value for r. Use the value 
of r that you find to solve the derivative equation for dr/dt. As a 
check on your work, since the radius is decreasing, your computed 
solution should be negative.

I hope that helps.

- Doctor Sam, The Math Forum
  http://mathforum.org/dr.math/