L4 Trigonometric Functions - Practice 1
Completion requirements
Unit 5
Applications of Derivatives
B. Related Rates Problems
Lesson 4: Trigonometric Functions
Practice
Once you feel confident with Related Rates: Trigonometric Functions, 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.
Click here for a formula sheet for Unit 5.
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.
Click here for a formula sheet for Unit 5.
1.
The beam of a lighthouse sweeps across the path of a boat cruising parallel to the shoreline at a speed of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»30«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math».
If the boat is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» from the shore and stays within the beam of the light, at what rate is the beam revolving, in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»rad«/mi»«mo»/«/mo»«mi
mathvariant=¨normal¨»h«/mi»«/math», when the boat has sailed «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» from a point opposite the lighthouse?
2.
A ladder «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» long is resting against the vertical wall of a house. If the top of the ladder is sliding down the wall and the angle the ladder makes with
the ground is decreasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», how fast is the ladder sliding down the wall when the
angle between the ladder and the ground is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/math»?
3.
Two sides of a triangle measure «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math». The angle between the two sides is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»50«/mn»«/mfrac»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math». Determine the rate at which the length of the third side is increasing when the angle between the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» sides is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/math».
1.
The beam of a lighthouse sweeps across the path of a boat cruising parallel to the shoreline at a speed of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»30«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math».
If the boat is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» from the shore and stays within the beam of the light, at what rate is the beam revolving, in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»rad«/mi»«mo»/«/mo»«mi
mathvariant=¨normal¨»h«/mi»«/math», when the boat has sailed «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» from a point opposite the lighthouse?
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle swept out by the beam.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance the boat has sailed from its initial position, opposite the lighthouse.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» be the distance from the boat to the lighthouse.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle swept out by the beam.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the distance the boat has sailed from its initial position, opposite the lighthouse.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» be the distance from the boat to the lighthouse.
Step 2:
State the given and required related rates.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«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»4«/mn»«/mrow»«/msub»«mo»=«/mo»«mn»30«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/msub»«mo»=«/mo»«mi
mathvariant=¨normal¨»?«/mi»«/math»
Step 3:
Write an equation.
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math» before proceeding.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»tan«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»x«/mi»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mi»tan«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mi»x«/mi»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»sec«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/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»«mfrac»«mn»1«/mn»«mrow»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/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»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/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»«mo»§#8729;«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»§#8729;«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«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»
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math» before proceeding.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»4«/mn»«/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»«mn»2«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mn»4«/mn»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mo»+«/mo»«mn»16«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»z«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msqrt»«mn»20«/mn»«/msqrt»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mi»cos«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»2«/mn»«msqrt»«mn»20«/mn»«/msqrt»«/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»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math»
into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
The beam is revolving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«mi»rad«/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»«mo»§#952;«/mo»«/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»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mo»§#952;«/mo»«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»«msup»«mfenced»«mfrac»«mn»2«/mn»«msqrt»«mn»20«/mn»«/msqrt»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«mfenced»«mn»30«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mfrac»«mn»4«/mn»«mn»20«/mn»«/mfrac»«/mfenced»«mfenced»«mn»30«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mfenced»«mfrac»«mn»1«/mn»«mn»5«/mn»«/mfrac»«/mfenced»«mfenced»«mn»30«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/math»
The beam is revolving at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math».
2.
A ladder «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»8«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» long is resting against the vertical wall of a house. If the top of the ladder is sliding down the wall and the angle the ladder makes with
the ground is decreasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math», how fast is the ladder sliding down the wall
when the angle between the ladder and the ground is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«/math»?
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle the ladder makes with the ground.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the height the ladder reaches up the wall.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle the ladder makes with the ground.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the height the ladder reaches up the wall.
Step 2:
State the given and required related rates.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mi
mathvariant=¨normal¨»?«/mi»«/math»
| Note the rate «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math» is shown as a negative since the angle is decreasing. |
Step 3:
Write an equation.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»sin«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»x«/mi»«mn»8«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mi»sin«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mi»x«/mi»«mn»8«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»cos«/mi»«mo»§#952;«/mo»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»8«/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»
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math» before proceeding.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«/math».
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«msqrt»«mn»2«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/math»
| Recall from the unit circle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«msqrt»«mn»2«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/math». |
Step 4:
Substitute the values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math», «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math», and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cos«/mi»«mo»§#952;«/mo»«/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 ladder is sliding down the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»2«/mn»«/msqrt»«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»x«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#8729;«/mo»«mi»cos«/mi»«mo»§#952;«/mo»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mfenced»«mfrac»«msqrt»«mn»2«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mfenced»«mfenced»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«msqrt»«mn»2«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/math»
The ladder is sliding down the wall at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»2«/mn»«/msqrt»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» .
3.
Two sides of a triangle measure «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math». The angle between the two sides is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»50«/mn»«/mfrac»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math». Determine the rate at which the length of the third side is increasing when the angle between the «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» sides is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/math».
Step 1:
Draw and label a diagram.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» be the length of the third side of the triangle and let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle between the sides of fixed length.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» be the length of the third side of the triangle and let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#952;«/mo»«/math» be the angle between the sides of fixed length.
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»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»50«/mn»«/mfrac»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»rad«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«msub»«mfenced open=¨¨ close=¨|¨»«mfrac»«mrow»«mi»d«/mi»«mi»a«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mfenced»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/msub»«mo»=«/mo»«mi mathvariant=¨normal¨»?«/mi»«/mtd»«/mtr»«/mtable»«/math»
Step 3:
Write an equation.
Since this triangle is an oblique triangle, the cosine law must be applied.
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» before proceeding.
Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is a length, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»=«/mo»«mn»2«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/math».
Since this triangle is an oblique triangle, the cosine law must be applied.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mi»b«/mi»«mi»c«/mi»«mi»cos«/mi»«mo»§#952;«/mo»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»10«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mn»4«/mn»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mfenced»«mn»10«/mn»«/mfenced»«mfenced»«mn»4«/mn»«/mfenced»«mi»cos«/mi»«mo»§#952;«/mo»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»100«/mn»«mo»+«/mo»«mn»16«/mn»«mo»§#8722;«/mo»«mn»80«/mn»«mi»cos«/mi»«mo»§#952;«/mo»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»116«/mn»«mo»§#8722;«/mo»«mn»80«/mn»«mi»cos«/mi»«mo»§#952;«/mo»«/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»a«/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»116«/mn»«mo»§#8722;«/mo»«mn»80«/mn»«mi»cos«/mi»«mo»§#952;«/mo»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mi»a«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»a«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«mo»+«/mo»«mn»80«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»a«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»80«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mn»2«/mn»«mi»a«/mi»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»40«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«mi»a«/mi»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» before proceeding.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»b«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»c«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mi»b«/mi»«mi»c«/mi»«mi»cos«/mi»«mo»§#952;«/mo»«/mtd»«/mtr»«mtr»«mtd»«msup»«mi»a«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mn»10«/mn»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mn»4«/mn»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»2«/mn»«mfenced»«mn»10«/mn»«/mfenced»«mfenced»«mn»4«/mn»«/mfenced»«mi»cos«/mi»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»100«/mn»«mo»+«/mo»«mn»16«/mn»«mo»§#8722;«/mo»«mn»80«/mn»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»116«/mn»«mo»§#8722;«/mo»«mn»40«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»76«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»a«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#177;«/mo»«msqrt»«mn»76«/mn»«/msqrt»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#177;«/mo»«mn»2«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/math»
Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math» is a length, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»=«/mo»«mn»2«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/math».
Step 4:
Substitute the values of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#952;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»and«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»a«/mi»«/math» into the derivative and solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»a«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math».
The length of the third side is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»57«/mn»«/msqrt»«/mrow»«mn»95«/mn»«/mfrac»«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»a«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»40«/mn»«mi»sin«/mi»«mo»§#952;«/mo»«/mrow»«mi»a«/mi»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»§#952;«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»40«/mn»«mi»sin«/mi»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«mi mathvariant=¨normal¨»§#960;«/mi»«mn»3«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mn»3«/mn»«mn»50«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»120«/mn»«mfenced»«mstyle displaystyle=¨true¨»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mstyle»«/mfenced»«/mrow»«mrow»«mn»100«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»60«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«mrow»«mn»100«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«mrow»«mn»5«/mn»«msqrt»«mn»19«/mn»«/msqrt»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«msqrt»«mn»19«/mn»«/msqrt»«msqrt»«mn»19«/mn»«/msqrt»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»57«/mn»«/msqrt»«/mrow»«mn»95«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The length of the third side is increasing at a rate of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»57«/mn»«/msqrt»«/mrow»«mn»95«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».