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Π = pi

V = volume

h = height or depth

r = radius

Unlike derivative rules, there is no set rule for a related rates problem.  Instead, you have to look at the information you are given and decide what you are trying to find out.   However, there is a general pattern to solving a related rates problem (I will bold the important steps)  Let’s take a look at this problem:

Example 1:     Air is being pumped into a spherical balloon at a rate of 5 cm3/min.  Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.

First thing you need to know about related rates is that they are all related to time.  So the derivative will be taken with respect to time.

Solution

First, draw a sketch of what is going on:

First of all, to solve anything in math, we need an equation that relates all of this information.  We know the balloon is spherical, and we are trying to find out what the rate of change (aka derivative) of the radius of the balloon is when the diameter is 20 cm. 

Next, we need to know how to find the volume of a sphere.

Volume of a sphere is   V = (4/3)Πr³

Now list everything you know from reading the problem

V’ = 5 cm³/min (V’ = rate of change of the volume).

r  = 10cm

Now list what you want to know

r’ (the rate of change of the radius

Now, you might be tempted to find out the volume of the balloon right now and plug it in.  But if you did that, there would be no way to find out what r’ is.

This is where implicit differentiation comes in.  Take the derivative of both sides, treating V and r both as variables.

You get:

V’ = (4/3)Π3r²r’

Clean it up to get

V’ = 4Πr²r’

Now remember what we know.  We know V’, and we know r.  We are trying to find out r’.  Since you know 2 out of the three variables, just plug what you know and solve for r’.  Here it is: