Problem

The upper right-hand corner of a piece of paper, …

26:29

Question

Answered step-by-step

Problem 72 Hard Difficulty

Two vertical poles $ PQ $ and $ ST $ are secured by a rope $ PRS $ going from the top of the first pole to a point $ R $ on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when $ \theta_1 = \theta_2 $.

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Carson Merrill

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Watch More Solved Questions in Chapter 4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Video Transcript

So if this problem d is going to represent the total length of the rope, so that means it's going to equal PR, which is part of the rope and R s, which is the other part of the rope. And we need to write this in terms of the angles and the heights of the polls. So what we end up getting is CO c can't of photo one is going to equal p r over our Q Coast, seeking a fatal to is going to equal r s over S T P. Q. Times Coast seeking theta one is going to equal PR so we just multiply this number of saints. That's just peek here. Here you multiply this on both sides and then similarly, we'll get the Ste. Coast SIGINT data to equals or s. So now we have that d is equal to this right here. Plus this right here. So when we differentiate that we get the the big D over D theta one is going to equal negative PQ co sick and potato one co tangent potato one minus S. T. Cosi conceded to who Tangent data to defeated two over D 3 to 1. So then we continue and we need to find an expression for this right here. We don't want to have those two differentials. So the way that we do that, as we find, uh, do you think it too? By noting that the co tangent of potato one PQ times co tangent of data one equals cure and S t co tangent faded too equals R t. So then that helps us because now we have the cutie equals P Q co tangent Say to one class S t co tangent. See that too? We differentiate this and we are able to get now that defeated two D theater one is equal to P. Q Acoustic and squared. See the one over Esti Costa Rican squared feet too. So now we can simplify our original derivative and plug this guy and right here where we were trying to replace it before. So we do that and we can simplify and we want to set this derivative equal to zero. When we simplify, we end up getting zero equals a negative co tangent, say to one coast Sea can't theater too plus co tangent stated to co Sequent say to one, then, um, we can simplify these, and we can bring this over here. So what we end up getting is a co sign of data. One equals co sign Oh, 3 to 2. Since they are both acute angles, then this only means that those are the 3 to 1 is equal to faded too, since they're both acute. And therefore the shortest length of the rope occurs when photo one equal status to

Get More Help with this Textbook

James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups

Study with other students and unlock Numerade solutions for free.

Problem

The upper right-hand corner of a piece of paper, …

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Kayleah Tsai

Caleb Elmore

Michael Jacobsen

Joseph Lentino

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

James Stewart

Calculus: Early Transcendentals

Kayleah Tsai

Caleb Elmore

Michael Jacobsen

Joseph Lentino

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Kayleah Tsai

Caleb Elmore

Michael Jacobsen

Joseph Lentino

James StewartCalculus: Early Transcendentals

Kayleah Tsai

Caleb Elmore

Michael Jacobsen

Joseph Lentino

James Stewart

Calculus: Early Transcendentals