Unit 5

Applications of Derivatives

B. Related Rates Problems

Lesson 5: Related Motion

The derivative of a variable with respect to time, like «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math», can be interpreted in two ways: as a rate at which a length or distance changes with time (as in the previous Examples) or as the speed of a moving object. The following examples will focus on «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» as the speed of a moving object.

Example 2

A «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» ladder, leaning against a wall, begins to slide down the wall. How fast is the bottom of the ladder sliding away from the base of the wall when the top of the ladder reaches «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» up the wall and is sliding down the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math»?

Solution

Step 1:

Draw and label a diagram.

Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance, in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»m«/mi»«/math», between the foot of the ladder and the wall.

Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the distance, in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»m«/mi»«/math», that the top of the ladder reaches up the wall.

Step 2:

State the given and required related rates.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨»«mtr»«mtd»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«/mrow»«/msub»«mo»=«/mo»«mo»§#8722;«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mspace/»«/mtd»«/mtr»«mtr»«mtd»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«/mrow»«/msub»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»

Step 3:

Write an equation.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»10«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfenced»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mn»100«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mn»2«/mn»«mi»y«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mi»y«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»

Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» before proceeding.

Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«/math».

Step 4:

Substitute the values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»y«/mi»«/math», and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mi»y«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»8«/mn»«mfenced»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mo»+«/mo»«mn»6«/mn»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»8«/mn»«mfenced»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»12«/mn»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»12«/mn»«mn»8«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»

The ladder is sliding away from the base of the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».

Watch the video A Similar Triangles Question to see how to setup and solve a problem that involves similar triangles.

Watch the video A Similar Triangles Question.

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