Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

The top and bottom margins of a poster are each $…

04:36

Question

Answered step-by-step

Problem 34 Hard Difficulty

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is $ 30 ft $, find the dimensions of the window so that the greatest possible amount of light is admitted.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Linda Hand
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Linda Hand

Numerade Educator

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

More Answers

05:03

WZ

Wen Zheng

02:13

Amrita Bhasin

07:57

Linda Hand

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Related Topics

Derivatives

Differentiation

Volume

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Kristen Karbon

University of Michigan - Ann Arbor

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

0:00

A Norman window has the sh…

03:10

'A) Norman window has…

08:36

A Norman window has the sh…

05:50

A Norman window has the sh…

06:48

A Norman window has the sh…

05:14

A Norman window has the sh…

07:44

Optimal Window Dimensions …

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

Yeah, okay, here's a norman window, it's a rectangle with a half circle on the top and we know that the perimeter is 30 ft and we want to find the dimensions of the window that lead in the most light. So that means we want to maximize the area. So we need a formula for area of this. It will be easy with the area of a rectangle plus area about half a half a circle. But we've got to name it cleverly. So let's Let's call the radius here to be x radius of the semicircle. So that means this side is two x. And then let's call these sides. Why? Since the vertical? And then if the radius is X, we need the perimeter of the circum half the circumference of a circle. So uh circumference is two pi R of a circle. So in this case circumference is two pi X but we only have half a circle, so one PX Okay, so then the perimeter is going to be this. Why? Plus pi X. That's why plus pi X plus Y plus two X. Okay, so that's our constraint constraint, Y plus pi X plus Y plus two X. Has to Equal 30. Okay, or two Y plus two plus pi X equals 30. And then we're trying to maximize area. So a equals well it's a rectangle. So it's areas linked times with so two X. Y plus the area of half a circle. So remember the area of the circle is pi r squared. So in our case pi X squared and we need half of that. So half the area is half pi X square. Okay, So to find the maximum of anything, you take the derivative and you set it equal to zero and then you solve and then you answer the question Okay, But we have a problem here in our problem is we have too many variables. We have X, Y and X squared. So we need to get one of them out of there and that's what the constraint is for. Okay, so now you decide do you want to solve it for Y or do you want to solve it for X? And then after you do that, which one do you want to plug in? Well, I'm solving for Y. I get to Y equals 30 minus two plus pi X. and so why is 15 -2 Plus pipe X oops. Two plus pi x oops. Mhm. Two plus pi X over two. Mhm. So now my area formula is two times X Times Y. Which is 15 -2 plus pi X over two plus one half by X square. So if I multiply two extra here, I get 30 X plus two plus pi X squared plus one half pi X squared. So I have 30 X plus two X squared plus pi X squared plus one half pi X squared. So one plus a half. That's three halves. So now I have 30 x Plus two x squared plus three halves, pi X squared. All right, That's a so now I'm gonna take the derivative with respect to X. So it's 30 plus for X. I've made a mistake. Right here, That's 30 x two plus pi X squared. And so that is 30 X minus two X squared -909ared. So that is 30 X minus two X squared -1 X half pi X squared. Excuse me? The way I knew that I made a mistake is because I was going to get all positives here and I wasn't gonna be able to set it equal to zero. Alright. Here we go again. Sorry about that. All right. Going to take the derivative derivative of 30 x. 30 minus for X -1/2 pie times two X. So 30 -4 x -9 X equals zero. So 30 equals 4 -9 times X. 04 plus pi So x needs to be 30 over. Okay, sorry four plus pi And then remember we had why was Why was 15 -2 plus pi X over two. So why is 15 two plus pi Over two times 30/4 plus pi. Okay that two cancels with that and give you 15. So now it's 15 times 1 -2 plus pi Over 4-plus pie. I don't know why I'm going crooked there. Okay, so um that's 15. Going to get a common denominator four plus pi That's four plus pi minus parentheses. T two plus pi Over four Plus pipe. So 15 times to over four plus by or 30/4 plus one. Hi so the dimensions are this Okay. X. is 30/4 plus pi. So make the base 60/4 plus by and then make the verticals. Whatever. Why was I forgot already? Um 30/4 plus. Why four Plus by Army. Okay. If you don't like that, change into decimals. Okay so the trick was first you have to get a good constraint and you had before that you had give everything good names. I could have called the base X and then the radius would have been one half X. And then I would have a bunch more fractions and clearly that would not have been good. Okay, so name everything correctly, write your constraint. Figure out what you're going to maximize takes to take the derivative saturday equal to zero. Super fun time.

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.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
71
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
47
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Volume

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Kristen Karbon

University of Michigan - Ann Arbor

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

0:00

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus …

03:10

'A) Norman window has the shape of rectangle surmounted by semicircle Thus the…

08:36

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus …

05:50

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus…

06:48

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus…

05:14

A Norman window has the shape of a rectangle surmounted by a semicircle of diam…

07:44

Optimal Window Dimensions Norman window has the shape of a rectangle surmounted…
Additional Mathematics Questions

01:56

"I need this done please
The dot plots show the numbers of children…

01:21

'7th grade math 10 points
This hanger is in balance. There are two l…

02:37

'im not sure if its too small or not
est 1 2 0f 15
The figure bel…

01:01

'can you solve this problem?
BCD
8
In right triangle ABC, alti…

01:27

'Help someone!!!!!!!!!!!!!!!!!!!!!!!!!!
The number of hot dog buns i…

00:12

'I need help with this real quick ( 20 points) :)
What is the equati…

02:28

'I NEED HELP PLZ THANK U BESTIES MUAH MUAH MUAH
If we identify the …

03:23

'The walls of a farm silo form a hexagonal prism as shown. What is the …

03:34

'Answer plsss will give brainliest
Find the surface area of the hous…

00:27

'Help me please , will mark
Luna observed that in the last 12 issue…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started