❖ Optimization ❖
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- Опубліковано 23 жов 2024
- Optimization Problem: Minimizing the Perimeter of a Rectangular Pen
📐 Master Optimization in Calculus with a Rectangular Pen Problem! 📐
In this video, we tackle an interesting optimization problem: how to design a rectangular pen with an area of 1000 square meters while minimizing its perimeter. This classic problem illustrates how to use calculus to find optimal solutions under constraints.
What You’ll Learn:
Understanding the Problem: Discover how we set up the problem with an objective function (the perimeter) and a constraint (the area).
Setting Up the Functions: Learn how to express the perimeter in terms of one variable using the area constraint.
Finding the Derivative: We’ll walk through the process of taking the derivative to find critical points and identify the minimum perimeter.
Interpreting the Results: Understand how to apply the results back to the original problem to find dimensions that minimize the perimeter while meeting the area requirement.
Why Watch This Video?
Ideal for Students: Perfect for high school and college students studying calculus and optimization techniques.
Clear Explanations: Follow along with step-by-step solutions that simplify complex concepts.
Real-World Applications: Learn how optimization is used in various fields, from architecture to farming.
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LIKE this video if you find it helpful!
SHARE with classmates or friends who want to improve their calculus skills!
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#Optimization #Calculus #RectangularPen #MinimizingPerimeter #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #DifferentialCalculus #AreaAndPerimeter #RealWorldMath #CriticalPoints
![How to Solve ANY Optimization Problem [Calc 1]](http://i.ytimg.com/vi/cUMvwG7wvzM/mqdefault.jpg)
Hi all! Please post comments, questions and anything else on your mind in the comment section! While I try to answer as many questions as I can, there are more questions than I can answer by myself so if you are someone who might be able to help, please do so! The community is so much better when everyone is involved and helping each other out. Our strength is our willingness to help each other!
Also please feel free to like or dislike the video. Your feedback is always appreciated so that hopefully I can make my future videos the best possible! Thanks and happy studies!
Hi! How did you calculate that y equals root square of 1000?
Where did you get the x^2? 3:41
Yes, this video is very good. However, I believe you shouldn't use an "x"(6:18 on the video) to represent multiplication. In video-wise, since there is narration, a person can clearly see which one you mean but on paper, without anyone talking, anyone looking at it is likely to assume that's a variable.
You seem to be skipping a few steps. I'm going to another channel to get a better explanation.
@@melissamartinez4727 the total area was 1000, and sqrt(1000) multiplied by sqrt(1000) equals 1000
I can spend hours poring over my textbook trying to learn this and get nowhere. But I come here and watch one seven minute video and understand it completely. You have a real talent, Patrick.
it's come to the point where i don't even do the homework or any practice problems, i just watch patrickjmt and do *fine* on the quizzes lol
Yeah. Math pretty much turns into do what you gotta do to pass.
same here, perfect evidence that the education system is FUCKED
But also some students are just lazy asf and just blame teachers when they're not doing all they can
You're the friggin best. Im taking my last math course of my entire life and it's just getting so bad, and every time I have any problem i just type the topic and JMT and here you are. It must be gratifying to be able to help such a vast number of people. You're FANTASTIC!
The ability to explain things simply is your special gift- Thank you
I find math really cool af at times. I like how it blows my mind but it fucks me real good cuz i suck at it :(
+Mearc well, I am not fishing for compliments here but I am really nothing special mathematically. Not at all. But, I work hard.
patrickJMT if you can be this good, then i guess i can be too ill work hard thanks
How would you solve this if x andy cannot be equal, it would help me in my homework.
sometimes working smart is better than working hard! just find the way that fit you buddy! best of luck
SAME.
ill just take the 0 on this part of the exam and cut my losses.
fear not homies, passed calc with a b- even while leaving this part of the exam completely blank lol
LOL
update: If yall cats have to take statistics, taking calculus first makes stats a WALK in the fucking park. Unless your prof is a douche canoe.
Chris Laffery you the realest nigga i've known. you probably a wshh troll
47-1 FloydMayweather #TMT
You have officially replaced my real math teacher. My scores on tests dramatically increased when I stumble on your channel! Thanks!
hi sleepy!
i am happy that my vids helped you out; it is knowing that they are making a difference for at least a few people out there is why i keep posting them.
congrats on doing well on your exam! if i ever make it to australia (would really loooove to go), you have to buy me a beer! :)
WHEN VISITING SOUTH AFRICA AS WELL..JUST TELL ME
Yes let us know when coming to south Africa
I would love to buy you a beer if ur still alive that is.
guess who isnt graduating
Did you graduate
Did you graduate?
@Gradient Soln-En oh um hi everyone !! yes, i did graduate luckily !! :)
Glad you did
Now I’m up on the chopping block
I just wanted to say thank you so much for your videos. I often find myself staring at a problem for hours, not knowing at all where to start, then I'll watch maybe half of your video on the topic and instantly understand the entire topic perfectly.
I have no idea why your videos are so effective, but holy hell you're a good teacher.
@patrickJMT What makes you so great as a teacher is that there is not a hint of sarcasm or condescending attitude or any sort of superiority attitude that so many math teachers have. You are the quintessential teacher, people love your unassuming approach and speech, I am a huge fan and recommend you every chance I get. You are a GREAT TEACHER!
Yup this and related rates suck. Everything else is easy as Pi
Dude fr
I agree augh I'm struggling with these
It all seemed pretty easy to me until I reached this and related rates, now I'm just beyond confused
@patrickJMT you deserve it, you teach exceedingly well. I made an algebra mistake the yesterday and asked my professor if the way i did the problem was correct and he said "i'm pissed that you don't know how to the algebra for this." your technique is concise and you're a great person for doing this, really. especially free of charge because most of us are broke as a joke.
I love that you can just mentally say, Patrick I need help, come here, type in the section at the top of the page, and get the lesson in very nice format. You have been helping me with math for 3 semesters now. thanks.
calculus exam today...after 16 weeks of lectures im finally understanding this stuff...thanks alot man...ur awesome...
@XtGu no problem. glad u like them.
The constraint idea really helped me set up these types of problems which to me is the hardest part, thanks a lot.
My BC Calc teacher showed us this video in class today. You, Sir are a genius... Whole class understands it now.
SMH i have a calc midterm in 2 hours and it's about optimization and related rates. pray for my dumb ass
How was it?
@@viv7169 lmao you ask her 2 years later. She probably got married and has kids now.
@@nahyan Still need to know haha. Left on a cliffhanger.
@@viv7169 well I have a math End of course exam happening tomorrow lol I can tell you how it goes
COME ON BISMA, WE NEED TO KNOW HOW YOU DID
Thank you. My current teacher isn't very good at explaining these concepts, but when you explain it, it seems simple.
first semester final for calculus ab coming up. thank you!
This was one of the exact problems on my homework, haha. I like how you came up with the constraint thing, puts it in perspective since I have a hard time setting them up.
I got a 89 on my AP Calc BC test and I used your videos since I was absent on some lessons! Thanks SO much!
seriously you are the best. like, i struggled through ap calc and am now taking calc in university and a friend told me about your videos. you are amazing! in high school i was able to get help from my friends but at my super competitive school people are a little more reluctant to help each other, haha. after a semester of watching your videos and my final being in two hours .. i actually kind of want to take calc2 since i have you helping me along the way. thanks again.
I think your videos are the only reason why I'm going to pass my AP Calc final.
of course! i prefer just using the first derivative test as that way i do not have to take another derivative. it is just a matter of taste as either the first derivative test or second derivative test will work.
My final is tomorrow, your videos have helped me study and clear up what my teacher made complicated. Thank you very, very much.
You are literally THE MOST AMAZING TEACHER EVER!!!!! I can understand more in your 7 min video than I can understand in 90 min of class.
You are an amazing teacher, i've been using your videos for 3 years now and I'm in calculus now and they are still very helpful
Thank you so much. Years ago I used your calculus videos, and it helped me understand calculus.
You teach it way better than most college professors do.
I hope you encounter beautiful women to sit on your face, for you deserve it for helping so many people learn and understand math.
daily routine after school.... patrickJMT! thank you so much for all your videos!!!!
this patrick gentleman is quite possibly the best explainer of maths I have ever seen
these problems are the least fun in calculus next to approximation
really? i like them. i show applications.
I think everything in Calculus is fun compared to the mathematics we did before
not important
Linear approximation is so easy
a critical number is a value in the domain of f that makes f ' either equal to zero or undefined.
to find what makes a fraction undefined, you set the bottom equal to zero. if that x coordinate is in the domain of the original function, then it is considered a critical number
hey ! every time i look for a video for math21(differential calculus) lessons i always make sure that it's yours. YOU ARE EFFIN AWESOME.
i should say, either can work... so long as the second derivative evaluated at a critical number does not equal zero; in that case, you would need to resort back to the first derivative test, which is another reason why i like that one!
We just learned about Optimization. We had differentiated instruction today in class, and we were given challenges after he taught us what to do. He referred us to this video after the lesson - well done
@patrickJMT i have to agree, in high school, I wanted a career in math , but it was was a joke because it was so simple, but when i got to cal in college, it was such a burden because it is a bore to understand the professors. The only way i really learn cal was from pat with his style of teaching made it so much more simple to understand and much more motivating.
calculus as a junior in high school FTL. thanks for the help.
You are a better teacher than my current calculus professor, and that is saying something. Thank you so much for these videos! You are the best!
optimization was the one part of higher maths that just wouldn't go in my head and i have my prelim tomorrow; but finally i get it !!! THANK YOU
@3lizza a square is a rectangle
this was very helpful! these word problems are so hard
word problems are the death of me
I love this types of problems....➕➕🅰🅰🅰➕➕
🍀🍀😊😊😊🍀🍀
📖📖🙏🙏🙏📖📖
Is there anything you can't do?? I literally see what the topic is for the lesson in my online class and instantly youtube it and you ALWAYS have a video on it. thanks so much bro
I still hate this stuff.
same
This video helped me out 10 years ago. Just came back to say thanks
ssssssooooo ccuteeee!!!!!!!!!!!
Infact you can replace 1000 with any number Alpha and the answer would still be sq. root of alpha and that leads us to a generalization that minimization of a recatngles perimeter for any given area leads to a square. The interesting question is minimization of area for any given perimeter.... would lead to a circle? It's an interesting one to solve.
I always go here for help. No other website, youtuber, teacher, tutor can help me as much as you do... Thanks so much!
Thank you so much. I have to do more complicated problems, but its basically the same idea. Thanks for taking it step by step, my professor just does so much stuff in his head and its hard to keep up.
Well, everything my teacher said made no sense today. Thank you so much Patrick. All day in class I was just thinking, "ugh.. once I get home I'll just watch patrickJMT and get this instantly," and I was right!
It's very interesting that you decided to move x in the denominator and move it to the numerator by making it x^-1. This is a lot better than doing the quotient rule.
Dude once ive finished my degree and im no longer broke ill donate as much as i can. You're the real MVP i owe you more than my tutors
The critical values of the denominator are the points that make the function undefined. x=0 makes this function undefined because, as you said, you cannot divide by zero.
Very extremely helpful, I never really had optimization down and today I have my Calc final... wish me luck! Thanks so much again!
I have a calculus exam today, which will have optimization on it. All things are possible with Jesus Christ🙏🏻 And thank you so much for these awesome videos!
How was it
watching your videos to get ready for my exam tomorrow! you are awesome!
I LOVE HOW U RESUME EVERYTHING SO SMOOTHLY IN THE END
This made no sense when my professor explained it for two hours. After seven minutes of watching this video I get it perfectly. You must be a superhero of some sort to make math this easy.
my thing is how would someone know how to set up a problem as such?
Are you asking why there is a problem that requires an optimization approach or how someone thought to answer these sorts of problems in this way in the first place?
like I dont think most people would know to set up the problem like 2X + 2Y
Blackie Robinsin the perimeter equation is just adding all of the edges together. So in a rectangle it is 2 length + 2 width; the x and y just make it easier to work with algebraically
Blackie Robinsin That's math for you...You just need to keep practicing .
The more you practice,the more familiar the solutions are to you.but...
Ain't nobody got time for that *yawns*
you simply don't. you skip the question and move on. #realtalk.
Guys, this video was made on April 1st... This is blasphemy.
Thanks a ton Patrick for all these wonderful videos that you create. They are precise and concise yet with an adequate amount of detail (especially with the derivations). Keep up the good work! I'm going to recommend my students to watch your channel. You've helped me immensely with the concept of derivatives where I've made notes from your tutorial and haven't forgotten those concepts. 😊
Hi Patrick.
I'd just like to sincerely thank you for the wonderful range of maths videos. I'm studying engineering at uni over in Australia... Was finding a few concepts hard to grasp in lectures, but they way you explain things makes everything seem a lot more clear. I managed to get 100% in a final exam of a subject with a 35% failure rate, largely due to your explanations ironing out the kinks I had in my understanding.
So basically, thank you for explaining things better than my lecturer.
I was struggling so bad with these types of problems until I stumbled onto here and everything started to make sense :) (THANK YOUUUUUUUU)
13 years later and it really helped
10 minute time limit. Damn does that bring me back.
This video made optimisation much clearer and my exam is today, so thank you a million times over!
Thank you, was stuck on the derivative for a while but this video helped
You forgot to mention that another critical point is also -sqrt(1000), because when you take the square root of both sides a negative number is also possible. It didn't matter at the end though, because a negative width is impossible, it was just something that I noticed you forgot about. Great video anyways.
yes because when you are finding areas of real world objects they happen to have negative area and space around them.
Sir Klutzy that's not the point being made here
Random Stuff he could have simply crossed it out just like how x=0 is crossed out but I'm sure it could have helped other students out by mentioning he negative square root
It's not a CP because it is not in the domain. You can't have a negative sqrt value, so it is excluded.
Anytime I have trouble I come here. You have helped me pass two exams!! Thank you!
Thanks for making this video because I would not have been able to finish my homework without it
You probably saved my math grade! Thanks a lot
+King Naveed I'm sure Patrick is just sick of hearing how much of a help he is -.-
U have helped me so much. I am currently a high school junior taking AP Calculus AB, and thanks to your help, I scored 95% on my midterms. God bless you, bro.
I'm planning to take AP Calculus BC next year. Hopefully you have videos on those topics as well.
@Charounson that is a good question
Amazing how this guy can explain optimization in like 5 steps while my teacher literally fills up both sides of the board on one question. Did what he said and not what my teacher said on a test today and got a C+ (MAJOR improvement for me lol). I'm coming back more often :)
thank you mister
you are a life saver for my calc test tomorrow
Patrick>Even the best math teachers
I don't know what I would do without your videos
It's 2020 and I'm watching. This is helpful. Thank you🥺🔥🔥🔥
@dexter127tep what kind of beans are in the can?
@baydood510 the answer is no you could not have because you have to make it so that you are not changing the number itself you see if you have 2x^2/x^2 then it winds up equaling 2 where if you have (2-2000)/x^2 when you separate it out you get 2/x^2 which is a number that does not belong in the equation and you would get a wrong answer. Hope that helps :)
You rock!!!
--From a very grateful math tutor who wants to brush up on her calculus
So he ruled out +/- squareroot 1000 because it can’t be negative when looking for length/height?
you are like superman, doing nice things for the world simply because its in your nature. if only the world had more people like you in other fields
@Maria Ahmed - You can't have a negative side of a rectangle.
It's 1:30 AM and I have a test on this & related rates tomorrow so it's time to watch every patrickJMT video on the subject and then cry myself to sleep
this makes learning this calculus crap SOOOO much easier! thank you!
Thank you so much. I had absolutely no idea what was going on until this. THANK YOU THANK YOU THANK YOU
@sarahgee1300 what would the max be? why must there be a solution to that?
I went to look for the Game Optimization, and end up with this.
Now I know how both works now. Thanks :)
@saltylemorse yes, in usa we stop using x for multiplication in middle school... i can not remember the last time i wrote 'x' for multiplication!
@ParkUniversityACM glad u think so : )
@skrevat yes it is nice to wake up each morning and to see lots of nice comments from people : )
it's a square 90% of the time
100% of the time, in this specific example
you have no idea how much i love you right now
@Master1906 I know that a length can;t be negative but mathematically the square root of 1,000 is (+-). If this was an AP test and someone wrote that x=(+-)the sq.root of 1,000 they would have gotten points off, thereby i feel that he should have mentioned that answer and said that it can only be positive.
Patrick, you're a boss. Thanks so much for posting , I would be studying for hours very morosely not understanding anything had i not discovered these videos.
My Calculus midterm is tomorrow. I started watching these a couple weeks ago and brought my test grades up 15 points. Let's hope the exam goes the same way.
I'm just grade four, and this is really easy for me!
Thanks patrickJMT for making it easy to understand! :D
Yeah, I got it! I made a few careless mistakes along the way, but upon watching I QUICKLY noticed those and came up with what you had!! A++++++