❖ Optimization ❖
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- Опубликовано: 28 сен 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!
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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
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.
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!
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
@dajakesta1234 @dajakesta1234 ha, not quite. they are already all here for free! of course, i guess i could take all the free videos down and charge for them (had offers from education companies to do this), but i do not want to do that. my intent is to create free content to help everyone, not just those with the $ to afford it. so, that is what i am doing : )
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
@dexter127tep what kind of beans are in the can?
@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!
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.
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!
excellent!
what part of texas you live in?!
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!
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.
first semester final for calculus ab coming up. thank you!
@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.
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.
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
10 minute time limit. Damn does that bring me back.
this patrick gentleman is quite possibly the best explainer of maths I have ever seen
My final is tomorrow, your videos have helped me study and clear up what my teacher made complicated. Thank you very, very much.
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.
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
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
Very extremely helpful, I never really had optimization down and today I have my Calc final... wish me luck! Thanks so much again!
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!
this was very helpful! these word problems are so hard
word problems are the death of me
I love this types of problems....➕➕🅰🅰🅰➕➕
🍀🍀😊😊😊🍀🍀
📖📖🙏🙏🙏📖📖
@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
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 :)
@Charounson that is a good question
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
Dear professeur may l ask you a question: optimisation is it a part of infinitésimal calculus ?
calculus as a junior in high school FTL. thanks for the help.
I was struggling so bad with these types of problems until I stumbled onto here and everything started to make sense :) (THANK YOUUUUUUUU)
@ParkUniversityACM glad u think so : )
Patrick>Even the best math teachers
@shelbieeex3 hope the test went well!
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 -.-
watching your videos to get ready for my exam tomorrow! you are awesome!
@Charounson For an integral to be meaningful, a variable (such as displacement ) must be changing. If you are talking about the integral of change in position with respect to time you are talking about the integral of speed which is path length. That is how far you travel along whatever path you follow, with respect to time.
I went to look for the Game Optimization, and end up with this.
Now I know how both works now. Thanks :)
So he ruled out +/- squareroot 1000 because it can’t be negative when looking for length/height?
Thank you, was stuck on the derivative for a while but this video helped
I don't know what I would do without your videos
thank you mister
you are a life saver for my calc test tomorrow
2:56-5:37
1. Speed Up
2. Play Star wars intro
+Martin the Malnourished Musket Man Tried it, now that the new star wars movie is out, and I'm glad to say I did well on the test. Thanks!
You could have took the square root of the area, and use that as dimensions. Ex square root 25 5, 5+ 5+5+5. Smaller then 25 and 1 for the perimeter.
why did you check x^2=0? 0 is not a critical point, is it?
1234whipcream No, only check the numerator.
when checking for critical points, you have to check for when the derivative is equal to 0 (namely the numerator = 0) and when the limit is undefined or doesn't exist (when the denominator is zero)
For written help on optimization, check out: fiverr.com/cheezees
this makes learning this calculus crap SOOOO much easier! thank you!
Why I'm attending Math lectures and we haveis Patrick Jmt
Thank you so much. My calculus professor is horriable, so bad in fact that the other professors even talk about him. Sadly, I'm going to have to ride it on through all the way to Calc IV. Oh well.
Thanks for the video man, it was great.
I wish I would have found your channel earlier. I'm looking at your vids now to review for the AP test tomorrow. :D
So if the constraint is the area and the perimeter has to be minimized...the answer is always just the square root of the area, yes?
What would you do if the question asked you to find a perimeter as large as possible?
I'm just grade four, and this is really easy for me!
Thanks patrickJMT for making it easy to understand! :D
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.
you're the reason why I'm not failing in my calculus class. thank you. really :)
SO, you use the critical point and find where it is decreasing and increasing to "prove" its the Minimum or Maximum point (most efficient in this example)?? Makes sense but seems unnecessary?
The square root of 1000 is 31.63. Usually calculators do not show answers with square roots that that why it came up as a decimal.
ap exam for calc tomorrow. good review videos, bro
You are one topnotch math guru! Thanks a million!
haHAA. 1st year struggling out here. I had an optimization problem on my calculus I exam that involved water filling up a cone or something and it was worth 10 points. I bent over
Thank you for being left handed, it really helps!
my pleasure, glad to help :)
Ok my main problem is that whether I'm maximizing or minimizing I get similar answers (such as x and y having the same values). How do I know if I'm doing it right?
how is y = sqrt(1000) if y = 1000/x, and x = sqrt(1000), wouldnt y = 1000 / sqrt(1000), which would be greater than a thousand? not making it a square?
Great Job! Very Clear, Thorough, and still a short video!
@Loggs777 wonderful : )
You are a godsend for the mathematically challenged!
The whole day I was saying to myself that I'll fail my tomorrow's test, but now after watching all your videos I don't think I'll fail it, even if I get a low grade :)
ONCE I GET A JOB I WILL GIVE YOU SO MUCH MONEY. YOU ARE A MATHEMATICAL GOD AMONG MEN.
Thanks sooo much! You have helped me get through my calculus class!
@trybalnet06 True... but in this case we know that length x can't be -sqrt(1000), so I think it's completely AWESOME to leave the negative sign out.
The way you left the negative sign out was AWESOME, Patrick! I had never seen it done so awesomely!
Anyway... thanks for helping me study for my upcoming finals! :)
hey Patrick, I got a question. If u take any number greater than square root of 1000, doesn't the P' get negative, not positive?
But we aren't looking to find area, we are looking for perimeter so wouldn't we plug that x back into our perimeter function to get our P?
You don't use the chain rule on 2000x to the negative first?
Amazing Patrick, you saved me on my calc test.
What if you had 2000 square cm instead of the side lengths? How would you figure that out?
i don't get why you got 2x^2? i know you multiplied across...but why? could you have left it (2-2000)/x^2 and still figured it out?
nice! i am only nerdy math geek by day... by night i turn into something cooler!
I have to be honest. I'm fascinated with the fact that he's left handed and has beautiful handwriting.
Wow, that was explained soo well, Thank You!!!!
You're very welcome!
Cramming for an exam tmrw, thanks for the help!
I don't get what you did to get rid of x^2 in the denominator after the step 2x^2- 2000/x^2
ok, i will keep you posted on that :)
Can you do minimal path problems PatrickJMT?
Would a square always give you the smallest perimeter and largest surface area?
yes, this is just showing you the techniques for more complicated problems.
what is it that you do differently when calculating the maximum vs minimum? thanks
you just made my day sir
If I ever become the architect I aspire to be, I'll name a bridge after you
no problem :)
at 4:23, what happened to the x^2 on the bottom of the derived equation?
CAN YOU PLEASE EXPLAIN TO ME WHERE CONSTRAINT CAME FROM...I GOT COMPLETELY LOST AT THAT POINT :(
listen to tony225!
this is the best math related comment i have seen thus far.
as a hard working student myself who appreciates knowledge (i still study cause i WANT to), he is 100% correct.
why do we check if it is a critical number or not?
sorry your teacher is not so great.
i am here 24/7
(s.root of 1000)*(1000/s.root of 1000)----the square roots canceling each other leaving just 1000. making it correct