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Optimization II - Cylinder in a Cone
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- čas přidán 1. 08. 2024
- In this video I will take you through a pretty classic optimization problem that any first year Calculus student should be familiar with. This is the second video on Optimization, if you want to look at a slightly less advanced setup check out the first here: • Optimization I - Class...
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Link to PDF of my work: drive.google.com/file/d/0B0x5...
You can also check out my podcast on iTunes at: itunes.apple.com/us/podcast/m...
Thank you very much sir. The x-y plane concept was very helpful!!
very practical to understand.we want more of such videos
Thank you sir, you've cleared my doubt!
very helpful
very helpful video
Good explain and easy to understand
yes
Thank you very much.
Thanks for this video sir!
You're very welcome!
Thank you very much sir
You're welcome- so glad you found it helpful. Be sure to check out my other vids and my math raps: czcams.com/play/PL59BEC8684FADD8D2.html
Thank you!!!!!!
This was the easiest way ive ever seen in all the videos that ive watched. Thanks a lot
Wow- thanks so much! Awesome to know it helped you out 😀
Thank you sir.
You’re very welcome 😀
So uhm... I dunno if this is gonna be seen, given that it's a 5 year old video, but first of all, thank you for the lesson! And secondly, should the height and radius of the cylinder always be the same or it's only for this case? Thank you!!!
Hey, thanks for watching 😄
In hindsight, that wasn't the best example to use for exactly this reason. I just happened to choose a radius that was half the height in this case. In general, the height of a cylinder or cone has no direct relationship to the radius 👍
How do you know to subtract 2r from 8 to get the height? I understand that since the cylinder is less than the height of the cone it'd be subtracted from 8, but why 2r? Also why do you do 8 divided by 4-r and not 8 divided by r?
There are a few ways to see it. In the video I initially just thought about the linear equation that describes the side of the cone… but if you keep watching from 3:35 I explain how to use similar triangles to see that the height is 8-2r
Do we get the minimum volume when r=0?
Sure, but that's not very interesting right? In general with the Max/Min problems one direction is interesting, and the other isn't (in math we call that trivial). For example, in this case it's interesting to see what the largest cylinder we can fit in the cone is, but not so interesting to see what the smallest is (zero volume). If on the other hand I had a certain amount of volume I needed to enclose, I may want to try to find the cylinder which will minimize the amount of material needed to enclose that volume.
Can you please clarify how you wrote linear equation h=8-2r in a jiffy :-) Curious to know the details behind the equation. Yes, understood the similar triangles part. Thank you.
Ok, so you can always get the equation from the similar triangles, but in this case there was a nice relationship if you imagine an x-y axis on top of the triangle. Since the height is 8, that’s the “y-intercept” of the line that matches the side of the triangle. Since the side of the triangle hits the “x-axis” at 4, the slope is -8/4 which simplifies to -2. Therefore using y=my+b the equation would be y = -2x +8, which I wrote as y= 8 - 2x because I don’t like negative signs after equal signs. Then I just replaced y with h to better match this specific problem. Does that make sense?
@@RealMikeDobbs Thank you Mike. Absolutely makes sense. I think you meant slope to be -8/4. Trying to brush up school math concepts for my son 🙂
Haha, you are absolutely correct; thanks for pointing that out, I edited the reply. Happy to help- many more videos on my channel. Thanks for watching 😄
How'd you get -2r?
It's just the slope of the line that represents the side of the cone. The y intercept is 8, and the slope is negative 2, so the equation is h = -2r + 8 or h = 8 - 2r. Here I am using h and r, instead of the usual x and y in y = mx + b. Hope that helps.
where did the 8/3 come from at 6:15
Solving the equation on the previous line. If the product is zero, then either (2pir) = 0, or (8-3r)=0. The second one is zero when r = 8/3
How did you make that animation in Geogebra?
Just create a slider, and then construct everything you want to animate in terms of the slider’s variable. Then, when you animate the slider, everything moves 😀
Sir hello
Uhmm what if theres a question
Express the height h of the cylinder as a function of r?
Hope.u notice sir😅
Hello and thanks for watching. In that case you would just solve the relation obtained from the similar triangles for r instead of for h.
@@RealMikeDobbs ty sir
How did you get critical point 8/3?
In order for 8 - 3r to be equal to zero, r must be 8/3. Just set it equal to zero and solve.
How do you get r³ sir?
Are you talking about 5:18 in the video? That’s just distributing and multiplying r^2 by r
How to find dimensions?
Hmmm.... I’m. It entirely sure what you mean. Can you be a little more specific?
A cone is MAYBE half the volume of a cylinder
1/2(b1 + 0) x height
base2 is 0 because the tip of the cone has an area of 0 or an area that is infinitely small. This may prove that a cone is half the volume of a cylinder. Let me know what you think!
In a way you're right. The formula you are referring to is the formula for the area of a triangle, and indeed, a triangle has half the area of a rectangle with the same base and height. You can tell that your formula is for an area because it contains only two dimension. You can think of a rectangle as a 2D cylinder, and a triangle as a 2D cone. Indeed, in 3D, a cone is exactly 1/3 the volume of a cylinder with the same base and height (in 3D the base is a 2D shape). In fact, this continues into all dimensions. In 4D for example, a 4D cone is 1/4 of the volume of a 4D cylinder (although this is impossible to visualize).
@@RealMikeDobbs so my formula can only calculate area for 2d shapes? I thought it could do the same for 3d shapes like a cube, cylinder etc
So here is my formula, and I'm sure it works for most 3d shapes.
Find the area of the 2 bases of a 3d shape, and in this case I'm going to use a cylinder for example, add the 2 bases up, multiply them by 1/2 or divide them by 2, and then multiply them by the height, which should give you the volume.
The same should be applicable to a cone in theory, because it has 2 bases, the tip being a base with an area of 0.
I'm an 8th grader and dont understand much about advanced math so please explain to me why or why not my volume equation is invalid.
Thank you
Wow- these are great questions for an 8th grader! Thanks so much for asking. You’re formula will work for what I call “stackable” 3D objects. I have a video on those if you want to check it out: czcams.com/video/gcDjF9Bo-0Y/video.html The short version is that your formula will only work for shapes where the two bases are the same size. So, you can actually simplify your formula a little bit. Since you are adding the areas of the two bases and then dividing by two, that will just get you back to where you started (since the bases are the same). So for stackable objects (area of base)(height) will get you volume. Cones are a different type of object which I call “pointy” objects. I have a video on those two where I get into a fair amount of detail on the whole “Why is a cone 1/3 the volume of a cylinder” thing. You should check that video out here: czcams.com/video/eBx6uqBY_Ug/video.html
@@RealMikeDobbs thanks for explaining!