Unit 5

Applications of Derivatives

B. Related Rates Problems

Lesson 3: Area and Volume

Area and Volume

Relations in mathematics represent a correspondence between two or more variables. This correspondence or rule is often expressed as an equation involving two or more variables. If one of the variables in the equation changes with time, any other quantity that depends on that same variable will also change with time.

For example, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mo»=«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mi»r«/mi»«/math» represents the relationship between the circumference of a circle and its radius. If the radius of a circle is increasing at a certain rate, what happens to the circumference of that circle? The circumference of the circle will also increase.
The rate of change of a variable implies a change with respect to time. If the circumference of a circle formula is differentiated with respect to time, there is a relationship between the rate of change of the radius and the rate of change of the circumference of the circle.

So, for the circumference of a circle, the radius and circumference are treated as implicit functions of time. As such, they must be differentiated implicitly with respect to time to determine the relationship between their rates of change.

According to the chain rule, if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math», then differentiating «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» with respect to time gives the following.

    
In the same manner, when differentiating «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mo»=«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mi»r«/mi»«/math» with respect to time, the result is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mi»C«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«mi»r«/mi»«/mrow»«/mfenced»«/math».

That is, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»C«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»2«/mn»«mo»§#960;«/mo»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»r«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».

The relationship indicates the rate of change of the circumference with respect to time is equal to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi mathvariant=¨normal¨»§#960;«/mi»«/math» times the rate of change of the radius with respect to time.

The circumference of a circle changing as the radius changes is an example of related rates.