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Afarmer'$ silo Is in the shape of a cylinder topped by & hemisphere: If the radius of the silo is 14 ft and the height ofthe cylindrical portion Is 44 ft, ...

Question

Afarmer'$ silo Is in the shape of a cylinder topped by & hemisphere: If the radius of the silo is 14 ft and the height ofthe cylindrical portion Is 44 ft, what Is the volume of the silo?Approximate volume (rounded to the tenths place) : 328583What Is the surface areaof the wall and _ tlame ofthe silo? (Do not include the bottom) Approxlinate SUIface area (rounded t0 %he tenths place}

Afarmer'$ silo Is in the shape of a cylinder topped by & hemisphere: If the radius of the silo is 14 ft and the height ofthe cylindrical portion Is 44 ft, what Is the volume of the silo? Approximate volume (rounded to the tenths place) : 328583 What Is the surface areaof the wall and _ tlame ofthe silo? (Do not include the bottom) Approxlinate SUIface area (rounded t0 %he tenths place}



Answers

A grain silo has the shape of a right circular cylinder surmounted by a hemisphere (see the accompanying figure). If the silo is to have a capacity of $504 \pi \mathrm{ft}^{3}$, find the radius and height of the silo that requires the least amount of material to construct. Hint: The volume of the silo is $\pi r^{2} h+\frac{2}{3} \pi r^{3},$ and the surface area (including the floor) is $\pi\left(3 r^{2}+2 r h\right)$

So figuring out volume is fairly straightforward. But when you start getting into you one of these, ah, the types of problems, then it can throw some people off. So the question I'm dealing with is a cylinder shape. So they got the silo and it's shaped like a cylinder, and it's 30 feet tall and it's 16 feet across. And so I'm gonna write. I know I'm being asked about volumes, so I'm gonna write my volume formula. That's the community use of a cylinder, which is the area of the base pi r squared times the height, and now I'm gonna identify the values that I have. I have the height, which is 30 feet. I mean, I'm gonna put this here, and I don't have the radius. I need the radius. I don't have it. I do have the diameter of 16 feet, and the radius is half the diameter. So if the diameter divided by two so I can say that that 16 divided by two, which is eight feet. So the radius is eight feet. Now I can substitute these values into my formula. So of the volume of a cylinder is high, are squared times the height and now I'm going to simply and resolve. So I have a volume. Ah, there. All right. So I am going to do my experiments first because that's the order of operations and eight squared is 64 tires 30 and then the volume is high times four times 30 is 1920. Yeah, so that's what the volumes is. The volume is 1920 high foods cute, but I don't want the whole volume. I want the volume of what's not being used. S o. What I need to dio is I need to find it says there's 1/4 a quarter of it is empty, so I need to find 1/4 or 1/4 of the total volume. So what I really want to do is I want to find 1/4 of 1920. Hi. Well, if and that's the empty volume empty part. Well, if I divide 100 if I divide 1920 by four, I get 480. So 480 pi ce cube is empty and it tells us to round to the nearest hundreds. So now I'm gonna multiply 480 times pi, and that's gonna give me 1507 0.96447 It doesn't look like a four for 73 And it it goes, um, on a little bit longer. And so when I round this, uh, there's just two different ways ripping around it. You can look directly here, Andi, cut it off and say, Look, I'm running to the hundreds, so I'm gonna look at the thousands place. So this is gonna be a 964. That's bullshit. And 160 than it is to 909 170. So this is gonna be 1507 and 96 hundreds. The other way that some people will sometimes round is that they'll look at the digits in ah, in like, a large picture on. Say Okay, well, if I look on the back here, then I actually have a situation where 96,447. That's closer to, um, 97,000. Because the seven runs the 4 to 5. This runs before to five. In this round. This is to a seven So you know, there's different schools of thought on this. It really depends on what your individual teacher is, then the require you to do. But either of these are extremely ending their rounded anyway because you're working with pie. So their approximate So the approximate empty portion is approximately 1507 point 97 or 96 Oh, yeah, keep it feet.

Okay, So for this problem, we have a grain silo made up of a cylinder with height 30 feet and a hem hemisphere here at the top. And together we know the volume is 15,000 cubic feet. And so the question is, what is the radius? Well, recall from the formulas that we know is the volume of a cylinder from the right as see why l equals pi r squared times the height. Okay, you know, the height is 30. Okay, In the volume of a hemisphere which is half of Ah, the volume of sphere is going to be 2/3. Pie are cute. And now why is it 2/3? Well, for a sphere, it would be 4/3. But since we only need the volume of half a sphere, 2/3 is half of fourth years. Okay, so now what we're gonna do is we know thes combined volumes. The cylinder volume, plus the hemisphere volume equals 15,000. OK, so we're going to set up our equation here, going to do it in blue. OK, so we have Let's see, we know we're gonna fill in h equals 30. Okay. So 30 hi R squared. OK, that's our cylinder. Plus, for our hemisphere 2/3 pie R cubed equals 15,000. Okay, now, the next step is just a little bit of algebra. Okay, So what we're gonna do is you notice We have, as I'm underlying and red, we have a pie term here and a pie term here. This isn't necessarily like necessary, but, um, it's gonna make some of your work easier. So what we're gonna do is we're gonna divide by pie. Okay? Remember, you have to divide everything by pie. Okay, So Heuser contents allow here. And what you're gonna let get left with over here is a decimal that we're going to relight down here. Okay, so now you have 30 r squared. Okay, I'm gonna write the 2/3 high. I'm sorry. The 2/3 are cubed over here just so we can get it an order of degree. That and then minus 15,000 buttery pie. So I'm kind of doing two steps and one here, so we're gonna take 15,000 divided by pi and subtract it from both sides. So you're gonna get over here. The 15,000 divided by pi and now all this equals zero. Okay? And then using your graphing device, uh, let's take a look at the graph once you plug this, um, polynomial in here Degree three polynomial, we're gonna look over here for the zeros and okay, so if you look, you notice on this. Sorry if you notice on this graph here, there are as I'm kind of Simon circling or underlining red. Excuse me. There are three zeros, but we got to choose which one makes the most sense. Well, these two over here are negative. It would not make sense in her life to have a negative radius. So we're gonna use this positive out over here, OK? And that is the value of your are Equalling 11.3 feet. Okay. And that so I'm boxing. Here in green is the radius of asylum.

The first step in this problem is to recognize the radius in the heights and substitute those values into the problem. The radius for this problem is three force, and the height is four inches. So let's stuff to to those values. And for R and H, we get to Hi and for Radius will put three force in and for haIf will put four plus. And now we have pi r squared and let's go to substitute the three force into the Are there as well. Pi times three Force squared. Now we'll grab our calculators and do some calculator work, starting with the first ah combination on the left side of the plus sign two times Pi times three force times four and I can plug that all into my calculator, multiplying all four items. When I do this, I should get 18.84. Then I'm gonna go ahead and multiply the right side of the plus sign pie and then times three Force times three force because 34 squared is 23 force multiplied together. So pi times three Force times three Force and I should get plus 1.766 25 In this case here, I will add these two items together. These two numbers and I should get the answer of 20.6062 five. This is my final answer.

Hello, everyone. The machinists are green. Cee Lo consist off cylindrical main section in a homicide vertical roof. If the Turin while him off, the zillo is 15,000 feet and the cylindrical party is 30 feet tall. What is the radius of the silo around you to the next 10 off the foot. First of all, let's draw a salute. This is the cylindrical part, and this is the Hamas physical bark. This is 30 feet high and the world news 15,000 feet, and we have to find the three D's. So first of all, the one you move hemisphere is the formula. Volume off hemisphere is two or three, but our cube and the volume of some indie areas Bye our square itch now Total volume off Zillow Equal toe B H plus Macy. So 15,000 equal to two or three by our cube plus by our square in the height is 30 feet square, 30 feet. So we have blood in the value off H, and it's not square. It's just 30. No need to simplify this equation. If it's simplified this equation, the secretion is going to give us to both religious plugging 22 over seven for pie, which is equal to 11 are cube bliss for 95 Our school negative. Suman it 750 equal to zero. No, What we need to do is solve this equation and find the roots off this equation. But if he used the rational zero Kerem, that will take a lot of time. So we will try solving it by plotting it. So let's just blot it first. Okay? Are square. Oh, sorry. I'm using the video. Give Alex instead of water here. Negative. 78750 Now we can see the draft. So here you can see there are three doors. One is this one. Is this in one? Is this so Let's look at the X values. They expel you is negative. Here they expel you is negative. So what we're trying to find is a physical quantity. That's radius and Grady's cannot have a value negative. So we will discard these two answers, and we will consider this one. So our reduces 11.279 So let's just write it down. And our unit is feet. So this is the answer, right?


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