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A cone-shaped drinking cup is made from a circular piece of paper of radius $ R $ by cutting out a sector and joining the edges $ CA $ and $ CB $. Find the maximum capacity of such a cup.

$\frac{2 \pi R^{3}}{9 \sqrt{3}}$

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Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

in this problem. It is given that a corn shape drinking cup is made from a circular piece of paper. Right? Of radius R. Right, So this is let us say this is a circular piece of paper here in Cuban. So a cone shaped drinking cup is me. Right? So now by cutting out the sector and this is the sector. Right? This is the sector. We have got out shade and joining the edges. C A N T V. That's right. Find the maximum capacity of such cuts. So how can we do? So we know that for the maximum capacity of a cup of whatever we are finding the maximum capacity we need to have we just differentiate the volume. Like cup will have some volume and for the maximum capacity it is differentiated with respect to it and it made it is made equals to zero. Okay, so this is what we need to do. So now to find the volume. All right, we need to find some values here. Right. So here we can find we can say r squared is equal to r squared plus b squared away because from by taking this to them. Yeah. Mhm. Thank so impeccable serum in right angle triangle hypothesis. Here are right, this is a Yeah. Okay, so now small are is quite equals to nothing but our risk wireless, it's a square. So now we know that for corn. Yeah, volume is what volume is one of 13 by odd square changed. Right, So here it will be one by three. Bye, I re squared minus its We have found out there develop a smaller square. So this will be multiplied my edge. Right. So now I have already told that for the maximum capacity Oscar up flood. Yeah, maximum capacity. Mm hmm Of cup D V. That is volume. You can see this is we we have found out already devi by th Is it was 20. So this should be the conditions. Right? So we need to do this. So this is very important. So now, yeah. Mhm here. Right. So when we will just differentiated we will get here D. V by the H is he questioned by us? Hmm bye mm. Today, I'm sorry this is pie are we square? It should be yes, it should be by our square By three because this is by square by three. Right? And it is that we are just differentiating with respect to add soap. Fire Squire, Squire by three is constantly. Okay, so this is a b minus again. We are just differentiating D 2nd part here. So we will get here what? This will be by in due today it's is squared right? Because the search is square and it becomes edge cubes. So that this way they will just rip the friendship this value three years right? The three Edge Square and all that other rallies and big constant for it because we are just differentiating with respect to aah so this year we will put this is equal to zero. So this will give us what this will give us bye are square equals two. Bye today. And to spare. Okay, so this would give us this is good. This is good. So this will give us at square is it was true. Honestly I like to do here date. Mhm. No, we know that our square is what we have already found. Note here this value are square as opposed to our square minus it is square. Right? So can we write the value of its here in terms of us so that we can find the value. I'm sorry this is our square. Okay, so you're We can write it like this. Any question one you can say about the questions are squares equals two iris quest minus edges square. That is artist red light today. This will give us good. Mhm. You just want to give me I respect what is your question to our inspired by today here? No. Really? Yeah. Right. So we have borne the value what is squared by today. So so here we have written somewhere. Right, So here we have written one more thing. Yes, this is volume. Right? So in terms of we will just try the value of our Squire -80 square. So therefore we can say I. D squared meditative square as well. All right. All that value in terms of as we will right here. Right, so This will be therefore we can say v. z equals two one by three by our square. That is our square. In terms of we can write two are inspired by today. Right, okay. And what is eight here? It could be it could be we can right from here. Right from here. We can. Right. Mhm. Are by Route three. So this will finally give us 2. 5 are killed by nine. Route three. So this is it. And so this is the maximum one. Oh God. So this is how we solve this problem. Isn't that easy? So I hope you understood. Thank you for watching. Mm.