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.
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.
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?
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.
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!
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
yes... with optimization problems like these, you always take the derivative of the optimization equation after you get it in terms of one variable. :)
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.
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!
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?
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!
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).
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
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.
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.
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
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.
Thank you so much!!!! I"ve been searching for a solution that was easy to understand for soo long:) was wondering if you are open to clearing doubts on other questions of geometry?
Kimberly rodrigues I'm so glad this finally made sense! Unfortunately I'm not available to help with specific problems, but I do have a short Geometry playlist on my channel.
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 :)
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)
In other optimization problems, we find critical points to prove what we've found is a maximum or minimum... do we not need to do that with this problem?
We should do it here as well. I already knew the critical point was going to work out, so I skipped testing it in this particular example, but you should always test it to make sure! 🤓
hello thanks for the video, need your help with question : need Volume intergral inside both the shpere X^2+Y^2+Z^2=2 and the Cylinder X^2 + Y^2=1...thank you
Hi why didnt u proof that when h=(2r)/sqrt(3) is the maximum/minimum point, and that we can assume that it is the maximum pt, since the derivative of dv/dh=0 is a stationary point and could be a maximum or minimum pt.
