Related Rates Of Change

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Suppose that you are blowing up a balloon, at a constant rate. How quickly does the radius of the balloon grow?

If an oil slick is gaining 100 gallons a minute, how quickly will it spread?

These are questions that depend upon related rates of change. One of the factors can be determined practically,( the volume of oil being added to the slick.) The other needs to be determined by connecting it to the first,( i.e. how does the volume of oil relate to the radius of the slick.

To consider the case of the oil slick, we first have to assume that the thickness of the slick is constant. The volume can then be directly converted to an area.

Let's say that the area is increasing at 500m²/minute. We can write this as

We now need to relate the area of the slick to it's radius. We can assume that it will grow as a circle, so

If we differentiate this we get

We want to find the rate of change of the radius. We can use the chain rule to find this.

You can see that the dA terms cancel each other out.

So we have an expression for the rate at which the oil slick grows, in terms of the current radius.

Could we get an expression in terms of time?

We would need to have an expression for r in terms of t.


	Divide and take the square root. Remove the t from the bracket, to make differentiation easier.
	Multiply by the power and reduce the power by 1. Take the terms into the sqr rt. Simplify
	We can check that our results are consistent, by substituting the expression for r, into the first result. After tidying up, you can see that the expression are the same.

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