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

Learn how to find the dimensions of a cylinder that will minimize its surface area. Since this is an optimization equation, draw a picture of the problem and write down what you know, identify optimization and constraint equations, and then solve the constraint equation for one of the variables so that you can plug your answer into the optimization equation and get the optimization equation in terms of one variable. Then take the derivative of the optimization equation, set it equal to zero, and solve for the variable. Then make sure you're answering the question you were actually asked.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[mrdjoy]

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

[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.

[orangutan]

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!

[garrettbalog]

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

[neiloconnor]

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.

[brookemaseda]

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

[kristakingmath]

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

[tammyhartel]

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

[nemovetinari]

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.

[nataliebasal]

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

[Baps]

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

[izzy]

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

[aayatzamil]

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

[kristakingmath]

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

[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.

[kristakingmath]

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

[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. :)

[charlesbrewer]

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.