By "rate" we mean the rate of change of some quantity (typically of a function, typically with respect to time). Often we have several quantities, each has its own rate of change. If these quantities are somehow related, then the rates of change must also be related (hence "related rates"), the question is how. The procedure for determining this is actually quite simple, we will explain it using a simple example.
Example: We are blowing up a balloon whose shape is always spherical.
We are blowing in the air at the constant rate of
Solution: Our experience says that the such a balloon initially grows fast, but then its growth slows down, so definitely the rate at which its radius is growing somehow indirectly depends on the current radius. We need to find out the precise nature of this dependence.
We start by identifying quantities, variables and data. The central parameter
is the radius r. However, it is also a function of time,
These are then the two rates we need to relate, the known dV by dt and the unknown dr by dt. In order to find a relationship between derivatives we always first try to find some relationship between the two quantities themselves. Here we have one:
Now we simply differentiate on both sides by time, keeping in mind that r is actually a function of time and therefore we have to use the chain rule.
We obtained a formula that relates the two rates of change, we can express the one we need, we will use the dot notation for derivative to emphasize the time as variable:
This formula shows that if we blow in the air at a constant rate, then the growth of radius slows down as the reciprocal of r2, so it slows down quite a bit. Back to the example, we need to substitute in the given data:
The algorithm we outlined above is summed up in Methods Survey - Applications along with an example, there is also an example in Solved Problems - Aplications.
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