Largest possible volume of a cylinder inscribed in a sphere (KristaKingMath)

[@TheStopts]

My teacher put a problem like this on the cumulative final. Of all the ways we could have demonstrated our knowledge of related rates, she chose this. Wasn't very pleased.

[@kristakingmath]

+TheStopts Cruel and unusual!

[@michaelmao1988]

This was hands down the best explanation I have found for ANY math problem whether it be in class, online, or tutoring...This was the first time in which the questions that popped up in my head were immediately answered. Thank you.

[@GameNerd346]

Her: *draws line connecting the radius to becoming a right triangle* My mind: *blown*

[@abby4537]

Clear, concise & error-free. Your final answer is equivalent to saying "The maximum volume of an inscribed cylinder is approximately 58% of the sphere's volume." That's plausible because visually, the cylinder in your diagram appears to be 58% as large as the sphere. As a follow-up, try answering this question: "At the point at which the cylinder is at maximum volume, what is the ratio of the height-of-the-cylinder to the diameter-of-the-sphere?" The answer is truly remarkable. Coincidence?

[@kristakingmath]

Glad you liked it! Thanks for letting me know! :)

[@KLaRue91]

Thank you so much! You and a couple other youtubers are the only reason why I'm passing Cal III so far this semester!!!

[@kristakingmath]

not necessarily. it's just the equation that constrains/limits the problem. basically it tells you what parameters you have to stick to.

[@monoman4083]

This is the kind of explanation I love. The detail is important to understanding. Keep them coming...

[@kristakingmath]

Will do!

[@sammanqueros8654]

I've spent so long looking for explanations on how we used Pythagoras theorem to solve this question, and you answered it for me so easily. Thank you so much!

[@DivineKiana]

This is the most beautifully explained math problem I have ever seen. Almost makes me not want to rip my hair out while doing my homework. Almost.

[@z51racer]

Thanks a ton! I wasted an hour trying to use (R, r - h/2) rather than (R, h/2) as my point on the sphere when I created my right triangle before I searched for help and found your video.

[@omargamal8]

The hardest thing in this problem is that to notice that triangle u constructed pretty smart :) Thx it's very helpful

[@cfcalle]

Thanks a lot! Spent a good couple hours fighting with this optimization

[@sugardust_kyu]

Fantastic video. Well-paced explanation with great clear visuals. Thanks a bunch. Might I suggest posting the problem: "Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius r" in the description? I think that would help people find this video. I was looking for this exact problem, but found it through a related video link.

[@Zyloly]

I have a calc 1 final in about 15hours and you just saved me a few marks...our teacher told us to know this example but I never understood how it worked till now..thank you :)

[@theresatruong4520]

You saved my life! I was so lost during lecture, but now I think I understand it better. Thank you so much!

[@nimasherpa9787]

Your videos are very helpful. I understood all the lessons that i didn't understand in class. Thank you :)....

[@kristakingmath]

Such a nice compliment! That means a lot! :)

[@Shoopdarupe]

Your virtual writing is better than my physical handwriting

[@周佩慈]

a student from Taiwan ,thanks your video ,it really helps me a lot !!

[@kristakingmath]

I'm so glad it helped! :)

[@kristakingmath]

i'm so glad they're helping, and i hope things are getting easier!! :D

[@enrichingexchanges]

This is very well-done! Thank you! Key insight is differentiating with respect to the variable in the optimizing equation, not the constant!!!

[@profarrington]

That was a very clear presentation of a difficult problem. Good job!

[@kristakingmath]

Of course! Good luck with the upcoming semester. Keep in touch and let me know how it's going. :)

[@kristakingmath]

yes... with optimization problems like these, you always take the derivative of the optimization equation after you get it in terms of one variable. :)

[@thedominator5729]

this question tease me so much but your explanation cleared my all doubts................... thank's a lot

[@MichaelMantion]

I love this video, my only concern and it is minor was when you aware reducing your final answer and cubed the square root of 3. I think some people might not understand that means 3 sqrt 3. Its perfectly clear to me, I love it, that is exactly what I would do, but I know some people struggle with roots. Keep up the great work. I do not understand why you have so few subscribers.

[@amberlee6752]

Excellent explanation! Very thorough, keep it up. In all honesty, this was one of the best explanation videos I have ever come across for math.

[@paperclipaos5947]

Thanks for clarifying, I was left a bit confused by that.

[@AdamMawby]

A perfect explanation. I was actually able to get the optimization equation all by myself! After that point, I ran into trouble when trying to take the derivative of optimization/volume equation because I hadn't considered *treating r as a constant*, which is why I had such trouble finding the derivative....But thanks to your video I was able to do so, and get the correct final answer!

[@syedarahman7381]

THANK YOU, CALCULUS SAVIOR. favorite calc help channel on youtube.

[@kristakingmath]

Aww thanks! I'm so glad I can help!! :D

[@samyyousef6566]

Hi there, so according to ur pleasant deduction we can say for the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius r we just calculate the sphere volume which is 4/3 * pi * r^3 and divide this vol by the root of 3

[@CameliaTex]

Excellent video! clearly explained and did not skipped the Algebra part! loved it! Thanx!

[@kristakingmath]

You're welcome, glad you liked it!

[@Mericali916]

By far the best part of these videos is, "In this particular problem..."

[@gowthambose5537]

you are dealing step by step so clear.. i like ur lectures..

[@kristakingmath]

Glad you like them! :)

[@gowthambose5537]

ty very much mam...

[@BradCozine]

Only 1 week until Calculus with Analytic Geometry 2 class... I'll be seeing a lot more of you soon. Thanks for being there!

[@AJITKUMAR-cz9bs]

I love your way of explanation ...!! Thank you so much mam.

[@kristakingmath]

You're so welcome! :D

[@mansnas7137]

You're excellent teacher. .you have done great job as well as we need.

[@kristakingmath]

Thanks!

[@kristakingmath]

So glad I could help! :)

[@kristakingmath]

I'm so glad, you're welcome!! :)

[@NerdOverPhysics]

Yeah!

[@kristakingmath]

Glad I could help!

[@tushainify]

awesome ! Everything is! the best part is the e-blackboard!

[@basiacdno8268]

That was super helpful! Perfectly concise and informative

[@kristakingmath]

Thanks, Barbara, I'm so glad it helped!! :D

[@spartanjon94]

your an excellent teacher, very thorough. thank you

[@kristakingmath]

thank you very much! :D

[@jamesdavies7040]

I don't usually comment on videos, but this was just so amazingly clear and easy to follow I had to say thank you!

[@kristakingmath]

James Davies You're welcome, I'm so glad it helped!!

[@kristakingmath]

You're so welcome!! :)

[@aedank8271]

thank you so much! I've been having a lot of problems with calc and your videos are a great help!

[@applelacquio8862]

This is very detailed, I learned so much. Thank you for sharing!

[@ostdog9385]

I like these videos because they show the relationship between both things and it's beautiful

[@rechardpreston5581]

I finally understand differential calculas becuase of you!!

[@nikhilbagh5518]

Your teaching is two good,,,tq so much for good explanation 😄

[@kristakingmath]

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

[@lightningjimmyjoejohnson8499]

Thank you. My test had this problem but the radius of the sphere was given.

[@erialettchill]

Such a nice flow in that explanation. Thank you!

[@kristakingmath]

You're welcome, I'm so glad you liked it! :)

[@johnlouelpaglumotan6861]

Thanks for the knowledge it makes me easier to find the greatest volume .

[@kristakingmath]

No problem, John! I'm happy to help! :D

[@kristakingmath]

Thanks!

[@MrKillerbunny1981]

Thanks, Watched a "Tested" video with Vsauce and Adam Savage and MIchael from Vsauce brought up the Napking Ring Problem, and i haven't been to school in awhile, so understanding a cylinder in a sphere first, then going back over the explanation it made more sense.

[@minhle-ey2ky]

OMG you saved my day. you are a great teacher. Thank you

[@ChristinaWahlquist-h5z]

You are a saint! Thank you so very much for the clear, concise explanation!

[@morganmitchell4017]

Thanks a bunch! This was incredibly useful.

[@kristakingmath]

Awesome! Thanks for letting me know!

[@milindjayantpatwardhan588]

Krista King Thanks for sharing ! It helped me a lot !!

[@kristakingmath]

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

[@Crazypandainacup]

Okay I understand everything, up to finding the V sub c. The answer you got was in terms of r cubed. What would the answer be just in terms of plain r?

[@kristakingmath]

good for you for working so far ahead, that's awesome! :) you'll definitely see the square root signs left in the denominator of a final answer, but i would recommend rationalizing it anyway. i know i left it here, but it's safer and cleaner to always rationalize. :) keep on learning, and good luck with the rest of algebra 2! :D

[@VoidShadowMoon]

Great tutorial for me!! You have just helped me to finish my assignment before the deadline! subbed :)

[@kristakingmath]

Done! That's helpful feedback, thank you for letting me know so I could try to make it easier for other people to find! :D

[@VishalGupta_1829]

Thank you for this superb explanation mam😃😃😃

[@kristakingmath]

oh no! but i'm glad you found this before going too much further, and i'm so glad it helped!! :D

[@kristakingmath]

i'm so glad it helped, and i really hope your final went well!! :D

[@smackdashitoutchu7755]

Great video. I especially like how you explained at 7:12 that r is treated as a constant rather than a variable because I struggle a lot deciding when letters are constants rather than variables (obvious constants like pi are easy but r in this case isnt, at least imo).

[@danayouuu]

i actually understood this. life saver.

[@TheVaiya]

Excellent teaching skills :)

[@mut57]

I don't know why this video popped up while searching for this problem but solved using lagrange multipliers. Good video though.

[@reinymichel]

Nice explanation. I would have defined the height of the cylinder as 2h, which would have avoided fractions.

[@niceboy171]

Excellent Video. I love the display format. It is very clean, and easy at which to look. Voice is very clear and sweet. I am impressed that so much expertise also comes in such a pretty package.

[@kristakingmath]

You're welcome! :)

[@deaner6421]

This is an old video but i just want to say I am amazed with that right triangle connection

[@paperclipaos5947]

I'm in algebra 2 and we are told that you are not allowed to have a radical denominator, later do we learn that it is possible to leave it as is, or would you still normally rationalize the denominator? (btw math is my passion which is why i know some integral calc)

[@kristakingmath]

so easy to forget, so i totally understand!! so glad it helped!! :D

[@srikanth621]

thank u so much the explanation is crystal clear....again thanks a lot...

[@mushtaqhussain8500]

Found it very helpful

[@MrCompenso]

Once again, you are amazing! The way you explain and describle the problems and solutions makes so much sense. I honestly wish I could make it up to you somehow, you've taught me so much. Much love to you!

[@kristakingmath]

+Grayfox772 Thank you so much, I'm glad you like the videos!

[@BigMarini]

I usually never comment on youtube, however, this was an excellent explanation of the problem.

[@kristakingmath]

Glad I could help! :D

[@kristakingmath]

Awesome! Glad I could help! :)

[@kristakingmath]

thanks! glad you thought so! :D

[@naghamhameed8828]

hey there. I'm Arabic and when I saw this vid about the applications of calculus ,it reminds me my maths book and subject and there's the same question in the vid in the book . I guess it will very useful and u r awesome in calculus 😻 keep it up ..💜

[@kristakingmath]

+Nagham Hameed Thank you so much, I'm glad you liked the video!

[@صفافارسبشارة]

Krista King | CalculusExpert.com u r welcome darling

[@صفافارسبشارة]

jolonar jojo this is my second account

[@joeyschmoey8182]

Greatest explanation ever!!!

[@kristakingmath]

Thanks, Joey! :)

[@kristakingmath]

Hopefully you can rock that section now. :)

[@SunilSundar]

Isn't the first derivative = 0 for both the maximum and the minimum? If you had to do the same problem, but find the smallest volume cylinder, what would you do differently? Thanks

[@bhaskardas8145]

Here we should check whether the volume is maximum for h=2r/√3. We can check it by 2nd derivative test.

[@kristakingmath]

Yep, it's always a good idea to double-check with the second derivative test. :)

[@amitayshahar]

does this mean the optimal dimensions for a cylinder are when the height and radius are in a sqrt(2):1 ratio?

[@hokhiyj]

this was very helpful Thank You

[@kbarry2295]

great video thank you so much. you are clear and get to the point, loved it!

[@benjomacho1757]

Please how to solve the bernoulli equation ?

[@carlpascual7211]

Thank you youre a great teacher

[@kristakingmath]

+Carl Pascual Thank you so much!

[@carlpascual7211]

+CalculusExpert.com keep making the world better one calculus problem at a time ☺😉

[@ML-mi7ns]

Super helpful! Thank you so much!

[@kristakingmath]

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

[@kristakingmath]

you're welcome!! :)

[@quenhueltehuantinnahuatl6930]

great explanation ! thank you

[@kristakingmath]

You're welcome, I'm so glad you liked it! :)

[@open5696]

Oh my.. Thank you very much!!! Wow just wow!! My instructor doesnt explain this very clearly.. And I learn from you.. :))

[@kristakingmath]

:D

[@Lexyvil]

Very informative video! Thank you! *Subscribed* It's so hard to memorize all of this.

[@kristakingmath]

Thank you for subscribing! It can be a lot, but practicing as much as you can will make you more comfortable and confident!