L5 Related Motion - Practice 1
Completion requirements
Unit 5
Applications of Derivatives
B. Related Rates Problems
Lesson 5: Related Motion
Practice
Once you feel confident with Related Rates: Related Motion, click on the Practice tab and complete problems 1 to 4. Check your answers by going to the Solutions tab.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.
1.
At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»:«/mo»«mn»00«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»a«/mi»«mo».«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«/math», a plane travelling east at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»100«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»500«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» west of a jet travelling south at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»200«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math». At what rate is the distance between them changing at noon?
2.
A «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» post is leaning against a vertical wall. If the top of the post begins to slide down the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», how fast is the bottom of the post sliding away from the wall when it is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» from the base of the wall?
3.
A trough has two ends, each in the shape of an equilateral triangle. The trough is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»9«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» long and it is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» wide at the top. Water is pumped in at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«msup»«mi mathvariant=¨normal¨»m«/mi»«mn»3«/mn»«/msup»«mo»/«/mo»«mi»min«/mi»«/math». How fast does the water level rise when the deepest point is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math»?
4.
If you are moving a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» tall vertical post away from a light source located «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» above the ground at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», at what rate is the end of the shadow of the post moving along the ground?
1.
At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»:«/mo»«mn»00«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»a«/mi»«mo».«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«/math», a plane travelling east at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»100«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»500«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» west of a jet travelling south at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»200«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math». At what rate is the distance between them changing at noon?
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance the airplane has travelled.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the distance the jet has travelled.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» be the distance between the airplane and the jet at noon.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance the airplane has travelled.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the distance the jet has travelled.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» be the distance between the airplane and the jet at noon.
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»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/msub»«mo»=«/mo»«mn»100«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/mtd»«/mtr»«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»t«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/msub»«mo»=«/mo»«mn»200«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/mtd»«/mtr»«mtr»«mtd»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»t«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/msub»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» after «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math», and then find the value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» before proceeding.
To calculate the distances «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» after «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«/math» hours, apply the distance formula «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»d«/mi»«mo»=«/mo»«mi»v«/mi»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mrow»«mn»500«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»250«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«msup»«mi»z«/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»«mrow»«mn»250«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mi»x«/mi»«mo»+«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mi»z«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«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»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»«/mtr»«mtr»«mtd»«mi»z«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»500«/mn»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«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»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»500«/mn»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«mo»+«/mo»«mi»x«/mi»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«mo»+«/mo»«mi»y«/mi»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»z«/mi»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» after «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math», and then find the value of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» before proceeding.
To calculate the distances «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» after «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«/math» hours, apply the distance formula «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»d«/mi»«mo»=«/mo»«mi»v«/mi»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left center center right center left¨»«mtr»«mtd»«mi»Find«/mi»«/mtd»«mtd»«mi»x«/mi»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mi»Find«/mi»«/mtd»«mtd»«mi»y«/mi»«/mtd»«mtd/»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»v«/mi»«mi»t«/mi»«/mtd»«mtd/»«mtd/»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»v«/mi»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»100«/mn»«mfenced»«mn»2«/mn»«/mfenced»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»200«/mn»«mfenced»«mn»2«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»200«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»400«/mn»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mrow»«mn»500«/mn»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mrow»«mn»500«/mn»«mo»§#8722;«/mo»«mn»200«/mn»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mn»400«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»300«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mn»400«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»90«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mo»+«/mo»«mn»160«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»250«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»z«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»500«/mn»«/mtd»«/mtr»«/mtable»«/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»«mo»,«/mo»«mo»§#160;«/mo»«mi»z«/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»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
At noon, the distance between the plane and the jet is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»100«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»500«/mn»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«mo»+«/mo»«mi»x«/mi»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«mo»+«/mo»«mi»y«/mi»«mo»§#8729;«/mo»«mstyle displaystyle=¨true¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mstyle»«/mrow»«mi»z«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»500«/mn»«mfenced»«mn»100«/mn»«/mfenced»«mo»+«/mo»«mn»200«/mn»«mfenced»«mn»100«/mn»«/mfenced»«mo»+«/mo»«mn»400«/mn»«mfenced»«mn»200«/mn»«/mfenced»«/mrow»«mn»500«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»50«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mn»000«/mn»«mo»+«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mn»000«/mn»«mo»+«/mo»«mn»80«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»500«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mrow»«mn»500«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»100«/mn»«/mtd»«/mtr»«/mtable»«/math»
At noon, the distance between the plane and the jet is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»100«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math».
2.
A «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» post is leaning against a vertical wall. If the top of the post begins to slide down the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», how fast is the bottom of the post sliding away from the wall when it is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» from the base of the wall?
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 bottom of the post 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», the post reaches up the wall.
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 bottom of the post 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», the post 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»x«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/msub»«mo»=«/mo»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/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»x«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/msub»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» before proceeding.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»2«/mn»«/math».
«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»3«/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»9«/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»y«/mi»«/math» before proceeding.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»2«/mn»«/math».
«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»3«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»3«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msqrt»«mn»5«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/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».
The post is sliding away from the base of the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msqrt»«mn»5«/mn»«/msqrt»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»56«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/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»2«/mn»«mfenced»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mo»+«/mo»«msqrt»«mn»5«/mn»«/msqrt»«mfenced»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/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»«mfrac»«msqrt»«mn»5«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/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»«msqrt»«mn»5«/mn»«/msqrt»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The post is sliding away from the base of the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msqrt»«mn»5«/mn»«/msqrt»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»56«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
3.
A trough has two ends, each in the shape of an equilateral triangle. The trough is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»9«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» long and it is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» wide at the top. Water is pumped in at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«msup»«mi mathvariant=¨normal¨»m«/mi»«mn»3«/mn»«/msup»«mo»/«/mo»«mi»min«/mi»«/math». How fast does the water level rise when the deepest point is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math»?
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math» be the height of the water in the trough.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»x«/mi»«/math» be the width of the water surface.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math» be the height of the water in the trough.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»x«/mi»«/math» be the width of the water surface.
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»V«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»h«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«msup»«mi mathvariant=¨normal¨»m«/mi»«mn»3«/mn»«/msup»«mo»/«/mo»«mi»min«/mi»«/mtd»«/mtr»«mtr»«mtd»«mspace/»«/mtd»«/mtr»«mtr»«mtd»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»h«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
Use the Pythagorean Theorem to find the height of the larger triangle.
Next, write an equation for the volume of water in the trough.
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
Use the Pythagorean Theorem to find the height of the larger triangle.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨»«mtr»«mtd»«msup»«mi»height«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mn»2«/mn»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«msup»«mn»1«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»height«/mi»«mo»=«/mo»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»1«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»h«/mi»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«msqrt»«mn»3«/mn»«/msqrt»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»h«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mi»h«/mi»«/mtd»«/mtr»«/mtable»«/math»
Next, write an equation for the volume of water in the trough.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»V«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»Area«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»triangular«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»end«/mi»«mo»§#8729;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»length«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mrow»«mn»2«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mi»h«/mi»«mo»§#8729;«/mo»«mfenced»«mn»9«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«mi»x«/mi»«mi»h«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«mfenced»«mrow»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mi»h«/mi»«/mrow»«/mfenced»«mi»h«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»9«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«msup»«mi»h«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«/mtable»«/math»
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»V«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»9«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«msup»«mi»h«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mi»V«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mfrac»«mn»9«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«msup»«mi»h«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»V«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»9«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mfenced»«mrow»«mn»2«/mn»«mi»h«/mi»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»18«/mn»«mi»h«/mi»«/mrow»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Step 4:
Substitute the values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»V«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math» into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
When the water depth is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»m«/mi»«/math», the water level is rising at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»3«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi»min«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»V«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»18«/mn»«mi»h«/mi»«/mrow»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»18«/mn»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mrow»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«mn»9«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»3«/mn»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
When the water depth is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»m«/mi»«/math», the water level is rising at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»3«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi»min«/mi»«/math».
4.
If you are moving a «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» tall vertical post away from a light source located «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» above the ground at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», at what rate is the end of the shadow of the post moving along the ground?
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance between the post and the light source.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» be the distance from the light source to the end of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mo»-«/mo»«mi»x«/mi»«/math» be the length of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance between the post and the light source.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» be the distance from the light source to the end of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mo»-«/mo»«mi»x«/mi»«/math» be the length of the post’s shadow.
Step 2:
State the given and required related rates.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«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»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mn»6«/mn»«mi»s«/mi»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»2«/mn»«mrow»«mi»s«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«mfenced»«mrow»«mi»s«/mi»«mo»§#8722;«/mo»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn mathvariant=¨italic¨»6«/mn»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/math»
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mi»s«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Step 4:
Substitute the value of «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» into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
The end of the shadow of the post is moving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The end of the shadow of the post is moving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
Find the rate at which the shadow is increasing in length.
Now, find the rate at which the tip of the post’s shadow is moving.
The speed at which the tip of the shadow is moving is found by taking the derivative of the relation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mi»s«/mi»«/math».
«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»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math», where «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» is the rate at which the tip of the post’s shadow is moving, «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» is the rate at which the post is being moved away from the light source, and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» is the rate at which the length of the post’s shadow is increasing.
The end of the shadow of the post is moving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» be the length of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance from the light source to the post.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the distance from the light source to the tip of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«/math» be the length of the post’s shadow.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance from the light source to the post.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the distance from the light source to the tip of the post’s shadow.
Step 2:
State the given and required related rates.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mspace/»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mi»s«/mi»«/math»
Use similar triangles to find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
Use the diagram from Step 1 to draw two similar triangles to represent the problem.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mn»6«/mn»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»s«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»2«/mn»«mi»s«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mi»s«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mi»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»s«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/math»
Differentiate with respect to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mi»s«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Step 4:
Substitute the value of «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» into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
The post’s shadow is increasing in length at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The post’s shadow is increasing in length at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
Now, find the rate at which the tip of the post’s shadow is moving.
The speed at which the tip of the shadow is moving is found by taking the derivative of the relation «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»x«/mi»«mo»+«/mo»«mi»s«/mi»«/math».
«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»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math», where «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» is the rate at which the tip of the post’s shadow is moving, «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» is the rate at which the post is being moved away from the light source, and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» is the rate at which the length of the post’s shadow is increasing.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The end of the shadow of the post is moving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»=«/mo»«mi mathvariant=¨normal¨» «/mi»«mn»2«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»25«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».