💧 Making a Rain Gutter 💧
ฝัง
- เผยแพร่เมื่อ 6 ก.พ. 2025
- ( • ❖ Optimization ❖ )
( • Max Area Enclosed by R... )
( • Optimization Problem: ... )
( • Minimizing the Area of... )
💧 Maximize Your Rain Gutter Design! 💧
In this video, we tackle Optimization Problem #3 where we design a rain gutter using a piece of metal that is 30 cm long. We divide the metal into three equal pieces and bend the end pieces upwards at an angle of theta to create a functional gutter that will hold water.
What You’ll Learn:
Understanding the Problem: We’ll explore the physical setup of the gutter and how it impacts water capacity.
Setting Up the Optimization: Learn how to express the volume of the gutter as a function of the angle theta.
Finding the Maximum Volume: Follow along as we apply calculus techniques to determine the optimal angle that maximizes water carrying capacity.
Why Watch This Video?
Ideal for Students: Perfect for high school and college students studying calculus and optimization.
Clear Explanations: Enjoy step-by-step guidance that simplifies complex concepts.
Real-World Applications: Discover how optimization is used in engineering and design.
📈 Don’t Forget to:
LIKE this video if you find it helpful!
SHARE with classmates or friends who want to learn about optimization problems!
SUBSCRIBE for more math tutorials, problem-solving strategies, and educational content!
Timestamps: 0:00 Introduction to the Rain Gutter Problem
#Optimization #Calculus #RainGutterDesign #Mathematics #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #RealWorldApplications #MaximizingVolume
Dude is so smart. Wish i could borrow his brain for tests.
I feel like my total cranial capacity is doubling for every ten minutes of watching PatrickJMT work these problems. Optimization and related rates problems seem to be about the toughest quagmires that derivative calculus has to offer.
ah hell nah bro fr said quagmire who else but quantum physics
I currently have a 57% in my calculus class because my professor is garbage and an unfair grader, and I need a 78% on my final to pass. I honestly feel confident in getting this score thanks to your videos. Much love :)
I can not thank you enough for these videos (and the rest of your calc 1 series for that matter). Between my best in person math teacher I've ever had and you on youtube, nothing in calculus I've experienced fazes me (yet).
Patrick, I owe my A in Calculus to you. Thank you for being my primary source of calculus expertise this semester. For whatever its worth, I sent you a 15 buck donation. You're the man! Keep up the excellent work.
I hope your making more money then a highschool teacher. Your videos are so easy to follow, so simple, and so helpful. I've learned more in 2 hours of watching your videos then i have from 10 classes of Calculus
In addition to solving the equation in terms of theta, you can also use the pythagorean thereom to find h or w in terms of the other variable, take the derivative with respect to whichever variable you chose to solve for, and then use that to find the optimal height or width. From there you can apply pythagorean thereom again to find your other variable, and then you could calculate theta by taking the arctangent of your (w/h) values.
glad i could help :) thanks a bunch for the donation!
@pimp2611 those were the old youtube rules. they have since changed.
This guy does a really good job! I just wish he would explain setting the problem up in more detail and finish the problem completely because the goal of the problem doesn't seem as realistic as it does in my math class. For me, it doesn't just end with the critical value, but more so a perimeter, area, volume, or a more extended answer. Regardless, these videos are really helpful. Keep it up!
Hey, Patrick I love all of your videos! Im just in pre-cal and I love optimization problems and this one blows my mind! Great job! Thanks for all of your wonderful lessons!
@urta93 no problem
it is one of the first trig identities that one learns; it is one of the pythagorean identities
thanx patrick i had the last 2 problems 4 my tuts u really helped me
I just got 100% on my mid-term thanks to you sir. Thanks from Canada!
i like how 2 of the three poblems you are doing are the ones done in my calculus lecture!
You really helped me through Calculus last year. Thanks!
He could not have explained this any better! This was on point! I'm seriously considering giving this guy $1 per month. Patrick, is there any way you could add a graph? Function A for 0 less than or equal to theta less than or equal to pi/2?
yea, i noticed that too! : )
subconscious 'up yours'? : )
or just an accident....
i wonder
I ended up using the trig identity cos^2(theta) - sin^2(theta) = cos(2*theta). You end up with cos(2*theta) = -cos(theta), which is 60 deg. The quadratic you ended up with is easier IMO, but this other approach worked out too.
You just saved my Calculus' final.I just saw all the optimization plus all the related rates videos. Totally worth it.
That's an interesting problem. Thanks for posting!. Pretty cool a bit of trig, calculus, trig identities. Great explanation :).
thank you so much for these videos, you have really helped me with my maths! way better than my maths teacher!!!!
You are incredible! We in Puerto Rico love u man!
This is definitely challenging, but man are you a good explainer! Thank you so much! Attempting to become a Materials engineer and you my man are helping me through Calc 1:)
Good luck man i am attempting electrical engeneering)
hi...great videos!
hmmm...i think you can avoid the derivation of trigonometric stuff by using the Pythagoras Theorem...just like:
total area=10w + wh
then build a triangle and by the definition of pythagoras theorem:
10^2= (w^2+h^2) and get the equation of "h"
now you can "plug in" it:
total area=10w + w([10^2+w^2]^1/2) and derivate it... once you get the solution of the equation you solve cos^-1(the solution)
0:05 the devil is in the math. Amen.
@patrickJMT Thank you again man, God bless.
0:44 tell me i'm not the only one that sees a face. :]
The piece of metal was given at the beginning of the problem to be a total of 30cm. And also given was the fact that the metal was to be bent into 3 equal parts. 3 x 10cm
Whoa. Impressive question! Never knew optimizations could be applied to angles
that's the more complex kind of optimization, there are also ones with cones and cylinders which are also very tricky
You are truly a Godsend. God Bless You. You should be teaching at an university.
man, I need a repeat of the cosine theta and sina theta's derivaives joh
this video got stuck and I have been trying to replay it over and over again but it still gets stuck when its at 7:57..i really wanna watch to till the end patrickJMT
For some reason, I read the title of this video as "Optimization Problem #3 - Making it Rain"
Am I the only one who thinks the drawing looks like a pair of angry eyes ?
Now that you mention it .___.
I cannot unsee this now.
I THOUGHT IT WAS RINNEGAD TURNS SHARINGAN HAHAHAHAHA
You have to think about the context. eg if you have profit, you wouldnt wish for your business to run at a loss. you'd want to find out where you'll get maximum profit. Same for cost of materials. if you want to build something, you'll look for the dimensions that use the least materials, right ? (because more materials = more money, which i'm sure given the current economic situation, you do not want. )
so in general, i'd say you should think "would this benefit me or would this be bad"
That rain gutter is staring at me with devil eyes.
Hey Patrick! In this problem we want to maximize the quantity of water i.e. the volume of the water hence how can maximising the area maximise the qty of water..since volume would be area into height of the gutter..
glad it helps everyone!!
also cos x = 1/2 at 5Pi/3 . Just wanted to let people know. So unless you are given an interval which does not include 5Pi/3 then you would normally list both these answers
When you're dealing with cosine or sign you should use exact values. Just think the numbers prettier he don't have to deal with any signs to the negative or driving sin. That's just what I did.
your videos are really usefull references, thanks a lot for your dedication
@1212naked i suggest a little shrine to start.... :)
Patrick JMT 2012 for president
10 minutes limit !!
I would give you all my time Pat , just keep going :")
This is EXACLTY what I need, omg thank you
thanks ill look into that. i must have missed that class
Whoops, one mistake. You would actually have to do 90 - arctan(w/h), since arctan(w/h) will give you the measure of the angle from the normal, in this case 30 degrees.
btw how did you come up with 3 10cm? is it in the given? have you considered other way of dividing the 30cm? just curious though.
as always thanks for the free vids! you surely help a lot specially me!
good math videos. By the way you sound almost exactly like Mr Vandreesen from Beavis and Butthead haha
i think that is a good way to use the videos ;)
Great video :D
I loved all of it!!
in 9:10 we can subtute cos(theta) by X the equation will became:
200*X^2+100X-100
-flips out the middle finger as a pointer- & -slips out a 'whoops' for reassurance of the otherwise case-
1:04 oops!
The cross-section is a trapezium. It's easier to use that and the formula for the area of a trapezium.
hahah the problems looks like an angry face screaming making a rain gutter !
Thank you sir for your video!
is it just me or does the picture look like a mischievous cat? XD
anyways, thanks a lot patrickJMT! ur videos helped a lot!
this will help with my exams the day after tomorrow
thanks gelly : )
that has to be one of the best comments ever :)
Awesome explanation!
Thank you!
Why do you not have to take the derivative of 100 in front of the cos or sin. Is it not a product rule?
Lmao when I heard the 10 time limit. Still saved me a decade later.
@ArrudA666 that is a restatement of the pythagorean theorem.
If you were to use product rule, the derivative of 100 = 0, so that would still just be 100*(the derivative of sin or cos).
How do you know when something is asking to be maximized or minimized?
Please make video of minimization for the same problem
very helpful videos. thank you
wait how can you ensure that its a maximum?
i mean couldnt it be a minimum?
dont you have to do a first derivative test?
@TheDuskMar I was thinking the same thing, saves a lot of time.
please tell me how to maximize the area of such a shrine
our AB Calc class went through Related Rates before going through this on a crappy textbook... nonetheless I didn't do well.
wait, if you already set the rectangle to a certain height, wont it be affected when you changes the angle? i mean, the flatter the angle, the less height.
the only part that confused me, which i guess i never learned, was how did you know that cos^2theta + sin^2theta =1 ? is that just common knowledge?
Because the 100 is a constant, not a variable. You don't use product rule when taking the derivative of 2x^2 because 2 isn't a variable. Same logic applies.
how can we check if it is the maximum or the minimum?
Couldn't you use Pythag to find the height in terms of width?
how do you know if the hypothenus of the triangle is 10?
Can someone help me, I worked this myself before watching and got the same derivative, 100cosx-100sin^2x+100cos^2x=0. Next I moved the -100sin^2x to the other side and divided by +100cos^2x, which gave me 100cosx=tan^2x. Why doesnt that work?
How do I solve it if there isn't a 10 value, just a variable a for each line?
Thanks for the video.
Thanks a lot! You are very helpfull!
I wish I have good memory, I keep forgetting Identities and all the important rules.
how do you do optimization problems without derivatives?
our class use this kind of problem but instead of that, we’ve used the area of trapezoid.
i think the only dislikes your videos get are from angry math teachers that are jealous of your awesome teaching abilities :)
finding the area of the trapezium would've also worked
all four teta angles are equal? how?
That is one angry Pokemon
I wish you were my calculus teacher !!
This stuff is so hard :(
It wouldn't be so bad if there were a standard or general way of doing each and every single one. :( :( :(
I have a test on this tomorrow!!! And related rates!!!!! :( :( :(
related rates for me is easier now this optmization is so hard for me
These videos are why I have a 97% in Calculus. :)
thanks a lot buddy, keep it real yo
what is the question? thank you!
@lilsamuraijoe oh yeah devil eyes! its scary as you thought it would be !
Oh my gosh I would of sworn the first thing I saw was a face. The angles are eyes the lines are the eyebrows and the swigging lines the mouth.