Problem

Consider the following problem: A box with an ope…

01:26

Question

Answered step-by-step

Problem 11 Medium Difficulty

Consider the following problem: A farmer with $ 750ft $ of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?
(a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it.
(b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols.
(c) Write and expression for the total area.
(d) Use the given information to write an equation that relates the variables.
(e) Use part (d) to write the total area as a function of one variable.
(f) Finish solving the problem and compare the answer with your estimate in part (a).

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 we're gonna start off with a formula where we can pick a value for width and get a value for the length of the other sides. So it's going to be 750 minus two W um, And since we have 750 ft of fencing to work with, we subtract two w from it to get what's left for the other sides. Um and then, since there are five parallel sides, what we have is that P is equal to 7. 50 minus two w over five. So we could alternatively, make a formula where we pick the length of the five parallel sides instead. But this is what our fencing is gonna look like. So, um, we have 200 ft here, 70 ft here, and then it will be broken up like this. Um, that would give us 70 times 200 which equals 14,000 ft squared. There's other drawings. You could do it there. Give us 9000, but experimenting with numbers. The max area we seem to get is somewhere around here. But we want to figure this out for ourselves. We want the absolute maximum. So for 11 B, part B. We're gonna let one side of the overall pen B X and the other side B y. So it's kind of like Lake. Yes. So this is going to be hard. Why? This will be X. And then there will be the rest of the pen and all of it subdivisions. Then we let it be the X width and the y length and a area so a is going to equal X times y. So then, for part that was C. So for part D, Um, what we have is letting x be the length of the end wall and y be the length of the dividing wall. What we end up having is two x plus three, uh, two x plus three x So this is the two X. And then there's the three acts right here. 123 and then we have two y right here and right here. So plus two y is going to echo five x plus two y and we know that five X plus two y is going to add up to equal +17 50 because that is our amount of fencing that we have. So now we can solve for y and we get y equals 750 minus five x over two. And we know that a equals x times y so we can substitute that in. So we get a equals X times this portion so we can just rewrite that as 375 x minus five house x squared. Then on top of that, um, for F, what we can do is differentiate this area and we end up getting a prime being equal to 375 minus five x. We set this equal to zero and see, um, that X equals 75. So what that tells us is that since when a prime of X is less than zero, you get when X is greater than 75 and a prime of X is greater than zero when X is less than 75. What we see is that X equals 75 is a local max. In that area, a of 75 is going to equal 14,062 750.5 ft squared

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

Consider the following problem: A box with an ope…

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

Catherine Ross

Heather Zimmers

Kristen Karbon

Joseph Lentino

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

James Stewart

Calculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Kristen Karbon

Joseph Lentino

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

Calculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Kristen Karbon

Joseph Lentino

James StewartCalculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Kristen Karbon

Joseph Lentino

James Stewart

Calculus: Early Transcendentals