solve-the-given-problems-a-conical-paper-cup-has-a-slant-height-of-100-mathrmcm-express-the-volume-3

Question

Solve the given problems.
A conical paper cup has a slant height of $10.0 \mathrm{cm}$. Express the volume $V$ of the cup in terms of $	heta / 2,$ where $	heta$ is the vertex angle at the bottom of the cup. Then find $d V / d 	heta$

Carson Merrill · Accepted Answer

Here we have a conical paper cup with a slant height of 10 cm to want to express the volume of the cup. In terms of the angle theta over to where data is the vertex angle at the bottom of the cup. And then we want to find DVD data so we know that the cup, if we look at the volume of a cone, the cup is going to be pi r squared H over three. So volume equals high, R squared Age over three. What we have to do in this problem is recognized at the height and the radius have a relationship. Um In terms of the angle measurement, if we have a high greater height and a smaller radius, then we have a much larger angle degree. But if we have a smaller height and a wider radius than that changes things, so we see that the radius and the height, this is the opposite and adjacent. So theta is equal to be inverse tangent or the arc tangent of the opposite, which is the height over the radius. So, by creating this relationship were able to rephrase the volume in terms of the value of data.

Arulmozhi T · Answer

you have political compass Lease from this open. API&#x27;s have people in three. DS sits in Geneva by cutting onto system, joining edges. So, Mom, we have to find the circumference. CIA field sleeper. We have a Peter on the base, my fine buddy. Three. So it is their conferences. Seven cents off. The blinker brings up. It&#x27;s nothing that the and company sector she Oh, but one of the biggest. We&#x27;re in his shifting piece of paper with six centimeter. We asked, You can&#x27;t get it. I didn&#x27;t feel just you, Which was s zit are the Children. There are six. It does vary party by three in anticipation. Five. It&#x27;s a box in a peak. Myrtle&#x27;s heard you one moment force into the guy. You know, we have to find the radius will stay open. L suppose this We have to find that we wonderland. It is so pumpkins? Yes, and I, if you can believe it. That&#x27;s you. Biggest hard and three and staff Prison moment off Also complained she basically that woman circumference of second. It&#x27;s a conference off this week. I How much do I owe? Who that is LJ written by What&#x27;s all this they get to artistic question. You know, obviously, country I&#x27;ve seen todo because this job you did. And now we have finally shake a stick up. I do? Yeah. You know, resuming. I just got something. We decided to try drivers the club through six into your job. So you&#x27;re saying goes all right? Oh, okay. We have and sprayer sleep too. That was hates way of yes, hours time. I thought you use President. She&#x27;ll be happy back in. Sword on it is five. It&#x27;s a lot of spice. Rare. The secret to the made here. Secular pick monitors six Fair catches a character 6. 10 Spy Rudolf six player. That is 36 minus 55. See the state of drugs Attention. Oh, yeah. Fortunately for Drew, it&#x27;s bring punks. You do said to me, Joe, now we have. Why didn&#x27;t the cook being the boy must hold. This cop is in the form of school. That&#x27;s a poacher. One by treat my aspirin, which and you know the values foot. All of it is from the tree. Goodbye. Our status five spring. You provide a street ball to do in central Italy. You This is a It is. It won&#x27;t in a fortune. I just cubits in village

Anthony Pisaturo · Answer

So this is probably section 8.1. Um, you were asked in a two sets of scenarios whether or not a rectangular coordinate system or a polar coordinates system would be more appropriate. So in part a, um we need to give directions to our house via a taxi driver. And so the taxi driver is gonna move along on this path, which is blue. And so in this case, it&#x27;s gonna be a rectangular coordinates that are gonna be more appropriate for the scenario and the reason this is because he can&#x27;t go the same path as the homing pigeon. He has to go. We&#x27;ll call this X units to the right And why units up on DSO In our case, we would have our we would tell him to go X units and why units and this would be more helpful in our scenario on and then in part B are poor coordinates would be more appropriate in the sun air scenario because the homing pigeon could take any path it once since it can fly over buildings. And so, in our case, we would tell it to go, uh, at a radius of well say, uh our distance and an angle of fada on, and so are more appropriate for Part B would be polar coordinates on, and that would be are.

Emily Burns · Answer

So we know that the circumference of the cone will be equal to this year&#x27;s conference of the paper. The concert operating our equations circumference equals two pi r multiplied by her angles over two pi. We&#x27;re just going to equal radius times air angle, which is going to equal six times fried pie over three, which then will equal Ted Hi. Just by filling in, this was our right Here is our original equation. And we simplify that because their two pies canceled out to leave us with this are times oh, you plug in our radius is six r angle is Fifi over three, which lose us with 10. Hi. So from there, I don&#x27;t want to get confused. That&#x27;s their answer. That&#x27;s just the start. So from R c equals two pi r. You bring 10 pie. We know there are is five for this one. We also So this 10 pie is our circumference of, uh, cone with the opening of the cup. And then this is our radius of the opening of our cup. The six is the radius of the paper. We&#x27;re just going to be a little bit bigger because you fold it up she learned from their weaken Dio That&#x27;s six squared. He goes h squared plus r squared. But he goes h squared Close that five squared because it&#x27;s our opening over cup reduce. You&#x27;re going to use our protagonist here. So you&#x27;ve got that h squared. He goes 36 minus 25 genes are h is going to equal the square root of 11. Therefore, to find our volume, you can do volume. He goes 1/3 i r squared age volume of a cone, which will be 1/3 pie. Five squared times are Hi. So then you&#x27;re going to do multiply that I your height, which is square root of 11. So you&#x27;re going to get about 80? Gonna get 86.8 June 9 rounded as your well, you

Ethan Somes · Answer

in this problem were given a circle of paper, and we&#x27;re imagining cutting a slit in it. So split, like right here, um, right here and overlapping the paper to make cold. Um, and we want to find the volume of this cone, given that the, um, overlap angle is 30 degrees. So first thing, I&#x27;m gonna do a strange that Drian because I like working in radiance. So 30 degrees, um, you can almost played by high over 1 80 degrees R degrees cancel and we get that is pi over six ratings. Okay, um, so then we know that then the overlap are clean, but we also know that this is four inches. The radius of that circle, the R Kling. Ah, this section here, um, is going to be our radius four times are, um, central angle pi over six, which gives us two pi over three. So this year is two pi over three. Um, we&#x27;re gonna use this to find, um, the circumference of the base of the cone. The first thing to notice the conference of the whole count. So the circumference of the earth of the whole circle. So the circumference, the whole circle. I&#x27;m gonna show pink, so that would be this whole. Okay, that is two pi r our, which is too high times a rate is four, which is eat pie. Okay. When we over left by that, um, orange amount by that two. Pi over three are circumference new. I&#x27;m gonna have see something. I was going to be eight. Hi. Minus that overlapping section. So you think about it this way. We have this point here is coming over. Becoming this pointers were scratching out those terrible cultures. This one year is coming over and connecting to that point there. Um, so our new circumference we ate Hi, minus two pi over three, which is 22 high over three. Um, OK, so knowing that we could find the new radius of the code. So now we&#x27;re picturing this code. It&#x27;s pretty flat cone. We know that circumference of that base. We&#x27;re finding new Vegas. So, you know, 22 pi over three equals two pi r. So if I divide both sides by two high, um, then I get the my new radius. Highest counsellor to use. Cancel is 11 3rd where radius equals 11 third&#x27;s. Okay, um So now we know a few things about this, um, coat. We know Radius is 11 third&#x27;s 11 3rd nelson of the slant height, Um, is forks? That was the original radius of the circle. We don&#x27;t know the height, so we&#x27;re gonna use the Pythagorean theorem trying the height. Um, so we can think of this as our as a right triangle here with 11 3rd and war that we&#x27;re looking for this I&#x27;m so been, say thanks. Squared plus 11 3rd squared equals four squared. Using our degrees here. That will give us, um X squared plus 13 point or that surrounded value equals sixteens. To track their 2.4, we get X critical. 2.56 Take the square root and xar height is going to be approximately 1.5 98. It is. Okay. Now that we have all of the values we need, we can use the volume of a code formula. So volume is high R squared times h over three, which will then be high times are radius, which 11 3rd squared times eight, which is 1.598 Continue to use the value saved in your calculator. I&#x27;m not the rounded volume. So we get a more accurate answer over three, and we solved that. We get 22.5 inches cute as the volume of our coat. Now, this was just the first part of the question. The second part was we&#x27;re looking for a generic formula Four. Any ankle help? Um, so instead of just 30 degrees, So we&#x27;re just gonna do the same steps before, except for using just this Alfa. So we&#x27;re going to assume person will assume this Alfa is in degrees because our original wasn&#x27;t to breathe, and I&#x27;m going to change it to, um, everyone Winter sumit in ratings. So I&#x27;m going to assume this IHS ingredients. It isn&#x27;t in radiance healing to do other stuff conversion. So we could say that this ark links yes, is four times outflow because our radius of our circle was still for, um and that&#x27;s the succumbing to the whole circle On the all use pink circumference of the whole circle is two pi r, which is still too high touch for which is still high on the circumference. The news or conference is going to be a pie minus for awful. He high. Okay, so then we can use this friend the radiant should be. So you know, eight Hi, minus four. Alfa equals two pi R For remember thinking about their cone? Um, some of the new radius there now, knowing the circumference. Um, we&#x27;re going tonight both sides by two pie you want I slit our and we get four minus to Alfa formed minus to Alfa over high equals R. Yeah. Um, so that is our radius. Now I&#x27;m gonna box that a little toe. Remind us. Eso Now we know when you tined with the heights of the Cohn is Yes, we know the radius. We know this land. I just for thinking we&#x27;re looking for that x so we can say four squared. Sure. Height equals from r squared plus x squares of 16 equals who were minus two Alfa over pie plus x squared. Um, so get 16 equals 16 plus four alpha squared over. Hi. Squared. Okay. Minus 16. Alfa over high plus X square was attract 16 from both sides. And I get, um, zero equals for squared over. Pi squared minus 16. Output per pie. Close excluded. I&#x27;m going. Teoh, Subtract this and we went to the other side. And so that will give me I&#x27;m 16 Alfa over pi minus for a squared over quiet squared equals X squared. So then therefore, X will just be the square root of $16 total pine minus four out of squared over high square. Um, okay, I&#x27;m still leave it in that form because then we can get our generic, um, for former volume when I was with box. That 12 volume equals high r squared times h over three. And we can just substitute those values. Nella, perhaps we got high are ours was four minus two. Alfa over high form leads to lawful Hi squared since there are times a two or three. So we got 16 Alfa over. Pie minus four squared over. Pi squared over three. So that&#x27;s our generic form. Assuming again that Alfa is a Brady

Cagri Selcuk · Answer

all right for this problem Problem. 34. We have a corn where the slant height is equal to three and we&#x27;re gonna try to figure out the other dimensions height and radius to produce the maximum maximum welding for this. Okay, first of all, if I would like Teoh set a relationship between height and radius, I can use the the presiding over a theory. So it&#x27;s gonna be like this. My first equation height squared, plus a radius squared is gonna be equal to the slant height squared, which is mine. And from there, uh, from there, if I sold secrets and for our So the radius squared is gonna be equal to nine minus height squared, and I&#x27;m gonna take the square that both sides so we get radius is gonna be equal to, uh, nine minus height squared under the square. It okay? And also, since we are would like to deal with the large possible volume So we know that the volume of the cone is 1/3 times pi times radius squared times fight. So in this case, when I see a radius, I&#x27;m going to substitute this expression into this evolving formula. So the volume is gonna be equal. 1/3 times pi times nine minus height squared Times heights. Okay, so let&#x27;s try to simplify this expression. Little bits. So we get 1/3 time supply times nine h minus h cute. And if I would like to find the maximum, uh, the greatest possible volume of this corn, I need to focus on this the prime h and said of the equal. Who&#x27;s it? Zero. So we&#x27;re going to get 1/3 pi times nine minus three height squared. Thank you equals zero. So if I simplify a little bit pie times three minus height squared is gonna be a colt zero. And from there, three minus high score, test vehicle zero. And from there, if if I sold situation for high that we&#x27;re gonna get height is gonna be able to Skurdal three. All right. If I would like to find the radius, I&#x27;m gonna be using this equation so the radius is gonna be equal to nine. Minus height squared, will be able to just threes and nine minus three under two square with And from there, if I simplify and radius is gonna be equal to just skirt of six. So with these dimensions, radius is equal to skirt of six, and height is equal to start of three. We&#x27;re going to We&#x27;re going to produce the largest possible volume for this corner where the slant height is equal to three.

Problem 60 Easy Difficulty

Solve the given problems.
A conical paper cup has a slant height of $10.0 \mathrm{cm}$. Express the volume $V$ of the cup in terms of $\theta / 2,$ where $\theta$ is the vertex angle at the bottom of the cup. Then find $d V / d \theta$

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