Dimensions that minimize the surface area of a cylinder (KristaKingMath)

[@mrdjoy9063]

I had to learn this for my engineering degree, and I just want to say that you explained this so beautifully. Amazing.

[@kristakingmath]

Thank you so much! Glad it was helpful! :)

[@kristakingmath]

You're so welcome! Glad you liked the video. :)

[@iluvmusical]

Thank you!!!! You just saved me so much! I was in a time crunch, and your video was concise, easy to understand, and visually appealing. You have my neverending gratitude.

[@neiloconnor5003]

Great explanation!! It only occurred to me recently that there ought to be a formula for minimizing the surface area of a cylinder, both to reduce construction costs, but also to limit radiation heat loss or gain. Moreover, the height to radius should be ratios, scalable to any size. I love geometry and trig, but never a big fan of calculus. Thanks for a fairly intuitive explanation. I was planning on using a spreadsheet for trial by error or range.

[@nataliebasal2781]

that good feeling you get after you plug the numbers back in to double check, and your answers are right!

[@kristakingmath]

I always found it helpful as well, so I do my best to include them. :)

[@orangutan879]

I know this is a late comment, but I had this type of question come up on my math homework and I don't remember the teacher explained this during class. I found your video and it was so helpful! Thank you so much!

[@kristakingmath]

You're so welcome, I'm so glad the video helped! :D

[@garrettbalog1699]

Still liking and replying to comments after 8 years. Amazing. Thank you for the help!!

[@kristakingmath]

You're welcome, Garrett! I'm so glad the videos are helping, and I'll always try to get back to as many people as I can! :)

[@brookemaseda657]

Thank you soooo much! This is the most helpful video I have ever watched in my entire life. I hope everything in your life goes right for you. Thank you!!!!!

[@kristakingmath]

Thank you so much, Brooke! I'm so glad the video helped! :D

[@kristakingmath]

The equation you find for A' is the equation that models how fast the surface area of the cylinder is changing as the radius changes. For example, if you want to know how fast the surface area is changing when the radius is equal to pi, you would plug pi in for r into the equation for A'. If the value you get back is positive, then this is how fast surface area is increasing at r=pi. If the value you get back is negative, then this is how fast surface area is decreasing at r=pi. :)

[@nemovetinari9063]

For those just rote learning that they should differentiate - here is little intuition: since the relationship between radius and area is parabola, dA/dr = 0 is either maximum or minimum. But the graph has no maximum - you can increase the radius up to infinity and the area will just keep on increasing. So this means when dA/dr is 0, we have the lowest point of the graph and that is the radius with lowest area. Simple yet genious application of calculus.

[@ismaelvbarry]

Clearly the best. Where were you when I needed you at the beginning of the semester? Thank you so much, your very helpful. Keep up the good work.

[@tammyhartel4690]

I fully understand now, thank you :D Feeling ready for my exam. This was the only chapter I struggled in

[@isaacsiegel1016]

i believe that DJ khaled once said about this page and I quote: "You Smart. You Real Smart. Matta fact baby you's a genius." #keystosuccess

[@kristakingmath]

i divided both sides by 4. since that gives me 0/4 on the left side, i just have 0. i get 4/4 on the right side, which just leaves me with 1. the r^2 disappears when you take the cube root of both sides. :)

[@martemaryjoy8967]

Why is the 2000 become 500?

[@kristakingmath]

You're welcome!

[@Baps1985]

Thank you for posting this. Your explanation was very clear and I was able to follow along so I could get my project done.

[@charlesbrewer6552]

This morning I was trying to remember how to work this problem out (I am in my sixties) and came across your video. Thank you. So:- Expanded out: r= (approx) 5.41926, h=(approx) 10.8385, d=2*r = (approx)10.8395. So maximum volumn for min surface occurs when height = the diameter.

[@izzy7159]

Thank you so much, also random but until watching this video I didn’t realize how nice it is to see a woman explain math. Idk it made me feel really happy!

[@kristakingmath]

You're welcome! So glad I could help. :D

[@280894280894]

Best video on this topic available on youtube. Many thanks!

[@kristakingmath]

Thank you very much!

[@kristakingmath]

That makes me so happy! Glad I can help! :)

[@kristakingmath]

You're welcome! :)

[@ChristinaWahlquist-h5z]

Thank you again!!!!!! May God Bless You!!

[@aayatzamil9605]

I had to learn this for my interior design course ❤️❤️

[@jimcar53]

Brilliant explanation - thanks

[@sheharyarakbar3447]

Mam your way of teaching is quite easy and effective..thank u

[@kristakingmath]

You're welcome, Sheharyar. :)

[@serbancojocaru3190]

Would also be interesting to show in the conclusion to the video that area is minimized when h=2r (when the height of the cylinder equals its diameter). Just an interesting observation.

[@blankblank5524]

Thank-you so much from Year 10 Aussie students.You are awesome,Krista!

[@jenniferxnicole]

So glad I found this video! Gonna ace my midterm this week now!

[@kristakingmath]

+Jennifer Stafford Awesome! Good luck on your midterm, I hope it goes great!!

[@jefferyb2253]

Thank you for this! Was just talking about this concept with my son (high school). He's excited about taking calculus soon. You're an excellent instructor 🙂👍

[@kristakingmath]

So glad it was helpful, Albatross! Good luck to your son on starting calculus! :)

[@Linhkinhbrods]

what does the can look like with the minimized surface area with its original volume. How is it gonna be different from the original shape of the unoptimized can?

[@kristakingmath]

Glad I could help! :)

[@samonge7]

you explain things wonderfully

[@kristakingmath]

Thanks!

[@ovatsugrosas]

Thank you very much for your video. I was able to help my nephew with his homework with your help :)

[@TomasUjhelyi]

I'm struggling to find what my optimization and constraint is. I've been told the cylinder has to have volume V and I need to know the shape of the can that is the most efficient cost and cost is equal to $1/per square cm. Would my optimization be the minimal cost of the can? Shape of the can is my constraint?

[@grenadegrega1303]

Wow, this is the exact video I needed. Thank you!

[@kristakingmath]

You're welcome! I'm glad it could help!

[@jp2kk2]

Great vid! Helped me conceptualize the process soooo much better!

[@kristakingmath]

I'm so glad it could help!

[@gabrielmonopoli7994]

Beutiful an excelente explanation. Whatbis the tool you are use , it is great

[@wheytomuchforher]

how could you dislike this video is the only puzzling thing about it... Thanks alot for the post it's always really helpful :D

[@kristakingmath]

Thanks! I'm so glad I could help!

[@eng-firas8178]

I love you and your lecture

[@EnjoiFood65]

What does A' represent? In terms of the cylinder? I'm writing my own world problem and I have given the Volume and radius. I followed all you did then plugged in my radius, but what does the number I contrived as A' represent?

[@kristakingmath]

I was here, you just didn't know about me yet, lol!! I'm so glad it's helping!! :D

[@adoperez15]

Your good thank you a lot first semester of calculus is the hardest. Atleast that's what people that took calculus in my High School say.

[@waitwhat7736]

omg you just helped me with my homework! tysm

[@littlept3837]

Absolute legend thank you for this

[@Ch4rmanderrr]

hi i have a question im confused at 7:09 how you went from 2000 to 500 cause im trying to substitute my question into your working if you get what i mean so right now im at (0= 4 pi r3- 750/ r2 ) sorry if this doesn't make sense

[@GhoulStabs]

This video is absolutely brilliant and taught me exactly what I needed to know! 👏

[@kristakingmath]

Oh good! I'm so glad it was helpful! :D

[@cb7708]

This is fantastic thank you so so much.

[@kristakingmath]

:D

[@mirandadewan9290]

very helpful video! now to find the height that would minimize the amount of material for the cylinder or can used would I just plug in the radius I got as the final answer into the height equation??

[@rubendarioanguloquemba6777]

Muchas Gracias, todo se puede lograr. Amo la Ciencia y la Tecnologia, Siento que todo se puede lograr con tus Clases. Gracias Maestra por su tiempo. ES POSIBLE. Matemáticas un gran misterio.

[@kristakingmath]

You're welcome, I'm so glad you're enjoying the videos!

[@mehirbanda7156]

this video was really helpful thanks Krista

[@kristakingmath]

Awesome! Glad it could help!

[@phoenixwings6441]

this... HELPED MEEEEEEEEEEEEEEEEEEE! I knew how to do the problem, but I didn't know why I did the problem. now I know WHY!

[@adriennekline5934]

If we check this with the second derivative test, we find that the test is negative (plugging in the critical point we found for 'r'), meaning this is not a minimum but a maximum?

[@margaretsarmiento2546]

Thank you so much! If I were to solve for the maximization of the surface area, would the procedure be the same?

[@tusandhu]

How do u solve for the max surface area

[@aboberto]

Great video!!! Thanks for your help ma'am

[@msnageshrao]

very easy explanation. Thanks a lot

[@kristakingmath]

You're welcome, nagesh, I'm glad it helped!

[@SolvingOptimizationProblems]

Interesting tutorial. Thank you very much! Keep it up.

[@kurafzogarak7974]

Thank you so much for your clear explanation. It helped me alot

[@kristakingmath]

+Faruk Karagoz You're welcome, I'm so glad it helped!

[@FJ2022]

Thank you. I know u did not forget to test test the value of r , but can u tell me why u did not test it? ^_^

[@callme_nandy]

Extremely helpful. Thanks.

[@charliecavanagh8217]

I was wondering what was the working out between the simplified area and area prime. how was the derivative formed from the equation before?

[@ThePinoyMamba]

Thanks

[@patrickdoll6036]

The exact video I was looking for. Thank you so much for your help! From Fargo, North Dakota (:

[@caseykramer2105]

Exactly what I needed too from Grand Forks, ND lol. #FightingSioux

[@Dborgz]

Thanks again, for another very helpful video! I have improved my grade and calculus skills because of you, you are awesome!! Keep it up!!!

[@melbri1793]

you helped me ace my calc final

[@kristakingmath]

For some, it certainly can be the most difficult. Glad I could help. :)

[@sweetdym1515]

This just helped me on my HW! Thank you so much!

[@kristakingmath]

Awesome!

[@billhill6795]

Love your videos! They seriously are a huge help...Thank you so much!

[@kristakingmath]

+Bill Hill You're welcome, I'm so glad they're helping!!

[@LostSoulAscension]

you are the best. Thank you so much, seriously.

[@kristakingmath]

You're welcome!!

[@jarrettwatson7555]

Thank you

[@kristakingmath]

You're welcome, Jarrett! :)

[@yassin1670]

That was so helpful! Thanks!

[@rossbeesley2575]

Hey, how would you calculate this for when the volume of the cylinder is an unknown value?

[@fabiolamaboga]

What if you were asked to maximize the cost of the metal? how would you do it?

[@kristakingmath]

Exactly the same way, except that when you use the first derivative test to investigate your critical point, your critical point will turn out to represent a maximum instead of a minimum. :)

[@saeedsaleh3909]

thank you so much

[@slaughterkill93]

This video was very helpful! Thank you!

[@patrickwiley4897]

Very helpful video! But are the measurements for radius and height in cm or inches? Doing a project that requires a literal construction of the container

[@kristakingmath]

+Patrick Wiley This problem is done in cm (metric units).

[@ankurwarrior9650]

Thanks a lot....its been a great help

[@kristakingmath]

You're welcome Ankur, I'm so glad it helped! :)

[@elenagaridis9392]

What is a constraint equation and how do you find it? Btw. great video!

[@kristakingmath]

the constraint equation is the equation that confines the optimization equation. for example, if i say that a farmer wants to build a animal pen with maximum area using 1000 yards of fencing, then he's constrained by the fact that he can only use 1000 yards, which tells you that the constraint equation will include the value 1000. :)

[@justinnanu4338]

Could someone recommend a video or website where I can learn how she found the derivative, A prime? Thanks!

[@syazwanasalehudin3895]

Thank you very much 😃

[@kristakingmath]

You're welcome, Syazwana! 😊

[@noobpeep2430]

This really helped thanks

[@kristakingmath]

You're welcome, Noob, I'm so glad it helped! :D

[@angelrai5993]

so how do you find the maximum if that's how you find the minimum dimensions?

[@sierrashahriar9611]

why does the square cancel out instead of the R? 4:50

[@fcccrlsn6498]

Using your solutions ..The ht is twice the radius ...h= 10.8 .,,,,.r= 5.4Will the height always be twice the radius when it asks for dimensions that minimize Surf area of a cylinder?

[@kristakingmath]

No, it won't always be that way. The height to radius ratio, or vice versa, can take on any value, not just this specific one. :)

[@jonahjohnson7230]

question: I am doing one right now thats the same question except the cylinder has no top. Would I simply subtract (pi)r^2 from the optimization equation or is there more to it then that?

[@kristakingmath]

Nope, it's as simple as subtracting (pi)r^2 from the surface area equation, like you said. :)

[@ganyu8804]

so what's the unit for h and r?

[@liamkandel3823]

thank you so much!

[@maxalawal5654]

thank u so much,,,But I have a question please ,What if my purpose was to find the Maximum surface?

[@bhowerter]

+Max alawal There is no maximum

[@noelmclolzorlor5464]

Hello, can you please explain why the exponent is -2? At 5:35

[@pebbles2681]

it's a basic principle of differentiation, you multiply the coefficient/constant by the exponent and subtract 1 from the exponent, and -1×2000=-2000 r^-1-1= -2

[@andreimatias4210]

Sorry for a noob question but i find it confusing to know if it is minimized or maximized. Isn't also maximized because the first derivative whether its max or min is 0? Or for those kinds of problems, is maximizing can't occur? Thank you for anyone who can answer

[@pebbles2681]

lmao 2 yrs later... to determine if it's max or min we do a second derivative and find the r-value, if the value is greater than 0 then it's minimum, if it's less than 0 then it's maximum

[@Rumiatingel]

8 years and it still relevant

[@Infusefps]

Math is math.

[@apeipz1766]

Really good video. Thanks!

[@ismaelvbarry]

Thanks. I'm sure that I got this answer right on my test. Are you a mathematics major and teacher? Or do you just love helping people in math?

[@cathrinachavez8934]

This is so accurate! Thank you so much 😊😊

[@kristakingmath]

You're welcome, Cathrina, I'm so glad it helped! :D

[@Jarratt95]

thanks babe just did my homework for me.

[@vanessaparada9655]

Honestly Calculus is tough. I'm a junior in high school currently in Calculus AB. I would have never thought it would have been this confusing. I was looking for a tutor and you seemed perfect :) , would you consider it?

[@kristakingmath]

I'm honored that you think so! :D Unfortunately, I don't have time right now to take on individual tutoring. But, I actually have an AP Calculus AB course in my Academy, and it might be a good fit! :) If you want to check out it out, there's more information here: integralcalc.pathwright.com/school/catalog/course/calculus-ab/

[@joabefernandes7286]

adorei :D preciso de uma professora dessa!

[@kristakingmath]

Just love helping. :)

[@zksanlin]

Thank you it really help..

[@kristakingmath]

Awesome! You're welcome. :)

[@andredevouissexavierii8970]

integralCALC yeah it sounds so darn easy, till the exam actually comes and kicks me right in the ass by surprise. When that happens i usually forget everything. my theory is, if i picture your face before the kick, ill probably retain what you said :D