5

Silo is built by putting cone of height and radius of 8 ft on top of a circular cylinder of height 20 ft and radius 8 ft_ An ice storm hits and covers the the roof ...

Question

Silo is built by putting cone of height and radius of 8 ft on top of a circular cylinder of height 20 ft and radius 8 ft_ An ice storm hits and covers the the roof of the silo with ice about 2 inches thick and covers the cylindrical portion about { inch thick: Use differentials to estimate the volume of ice covering the silo. Will our estimate be an overestimate O an underestimate?

silo is built by putting cone of height and radius of 8 ft on top of a circular cylinder of height 20 ft and radius 8 ft_ An ice storm hits and covers the the roof of the silo with ice about 2 inches thick and covers the cylindrical portion about { inch thick: Use differentials to estimate the volume of ice covering the silo. Will our estimate be an overestimate O an underestimate?



Answers

A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top (see the figure). The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height $h$ of the silo that will result in a capacity of $11,250 \pi \mathrm{ft}^{3}$.

So you know you want a silo that has a cylindrical base. Mhm. So it has a radius of our in height and we're not going to consider having the bottom said no bottom and we have a hemisphere up here and it said that our volume is fixed and we know the volume for this is going to come from pi r squared times age, that's the volume of the bottom. And plus we're going to have that hemisphere and so that is 2/3 pi r cubed. So that's the volume of this object. And again this number is supposed to be a constant. We don't know what it is, we're just going to call it call it be. And so I don't want my constant H two B in my equation. I want that just to end up being a variable are so we need to solve for H. And so if we subtract they two thirds pi r cubed from both sides and divide by pi r squared, we will end up getting H. Now I'm going to clean this up a little bit. I'll move this if I can I'm going to move this up a little bit. There we go, I'm going to split it up into the over pi r squared minus and then I'm going to simplify this down a little bit. These pies will cancel here and two of the ours will cancel here and we'll end up with two thirds are and that's what the height is equal to. So now we want to deal with the cost. We want to minimize this cost but we remember that this is what H is equivalent to in terms of our constant volume and other radius that's changing. And so we want to write a cost equation and we know that this top cost two times more, two times more cost than this, the sides of that cylindrical side. So I'll just read the cost equation all in terms of blue. So this isn't going to literally figure out the cost because we don't know how many how much a square foot cost or square yard, whatever, whatever square meter. We don't know what the unit is are going to use, but we'll just say the cost in terms of our is equal to and we'll do a relativity anyway. So we know that the volume of a sphere is four thirds pi R squared. And we know that the, excuse me, cubed can't get the tube to draw up here though, that's supposed to be cubed. Well they're supposed to be a three there and I can't get that to draw up there. We'll just put a three down here and that should be to the third power, so that's volume. And remember the surface area of a sphere is for pi R squared. So we have half of that. So but we also know that the cost is twice as much. So the the surface area of this to make that hemisphere is two pi R squared. But then we're going to have to pay twice as much for this as the other. So we're going to multiply it by that factor too. No, we need to find the surface area of this exterior and I think of chopping it here And laying it out, it becomes one big rectangle that has a circumference of two pi R times the height. But oh, I don't want height in terms of age, I want that in terms of and are. So let's do a substitution. So I've got four pi R squared here and I have two pi R times H. And I have to write down what H. Is H. Is this value here? V. Which is a constant number over pi R squared. So let's write that down the over pi r squared, I think I wrote that. Right, okay. And then minus two thirds are yeah, so let's get this cleaned up a little bit. We're gonna kind of just draw a line there and so this costs relative costs anyway is four pi R squared plus and let's distribute here. So when I distribute here that pie cancels that pie and that are cancels that one of those are. So we get to a V. Over so that two pi cancels with that pie that are cancels are one of those ours. And so we end up with two V over R. And then when we continue to distribute here uh nothing cancels there. We have four thirds pi R squared. Now We can combine these two together. Let's get a common denominator. So if I multiply this is over one and multiply top and bottom line three this becomes 12 over three. So I have the subtracting these two, I have 8/3 pi r squared. And then I have plus to the over our and remember this V is a constant number. I just don't happen to know what the volume is. Now we want to minimize that cost which means we want to find the derivative And then set it equal to zero. So we can use our power rule here in two times. This becomes 16/3 pie are plus. And then when we take the derivative here, we can also think of this term as to the times are to the -1. And remember this is a constant. And so when we do that, we get negative two V. And then are to the negative to power, which becomes over R squared. So there's our derivative. And we want that slope to equal zero. So now we just have to do some algebra to solve. So I'm going to leave this 16/3 pie are on this side and move this to be over our square over to this side. Multiply both sides by R squared. The net cancels those out. And then I want to multiply by this reciprocal. So I'll end up having r cubed equals to the times. And then this reciprocal this becomes 3/16 pie. And I can reduce that down. And so I get Taking the each side to the 1/3 power or cube rooting each side. I get R is equal to three v over eight pi to the 1/3 power. And that is going to minimize the cost. So that is the radius that will minimize the cost. And then I can come back over here and do a substitution to find out what that height is. And and do my substitution over here and find out that uh and let's see which form shall we use, we can use that. We'll use this form, we have V over and then we would substitute pie. And then right here for the R squared, we're going to substitute this in place. So we'll have three V over eight pi. And then this is all to the one third power, which is going to make that to the 2/3 power minus two thirds times three B over eight pi to the one third. And that's our H value. And now there's quite a bit of simplifying to do here and one half and I'm not going to go through and do all the algebra for this, I would leave that for you, but in the end we end up finding that the H value becomes this, and again it's gonna take me a little bit of work to do this. I would go through and uh you can do some rationalization uh to get some common denominators and do some changing here to get that in that form. But all in all it will end up being this. So this is what your height your radius should be and this is what your height should be mm.

To construct a silo. That is a hemisphere, um, you're just have to bear with my diagram. Here is Tasked Sphere, right? And then on the bottom is a cylinder again and again. I'm not a nerd. It so bear with me. But it's simple in your now cylinder has medicines are in the habit of your heads are all right. So it it theoretically go back like that. Make it here. So what it asks us to do. It tells us their volume is easy, bro. All right, that volume pride Two things comprised of the volume of its here. But it's half of this year. Um, 4/3 aren't you? Which is what, normal boy with tears and 1/2 of that dessert fire and then plot by our expert e Hey, that's the value. And then it tells me I want to minimize I was being cut off his basement surface area. I have two pi r square. That's cemetery of fear. Normally four tires for its here. Two fires pay for half of that and then two pi r h is my volume of my calendar. It tells me something doesn't cylinder is only half of car. So this is gonna act like I do that caught liked as much. Alright. But that means that they cost from chipotle become work shy aren't very plussed soup. I are a now I want to replace eight Do is going to use this formula. I'm gonna say then easier O minus 2/3 of pi r cubed, divine and by pi r squared it's evil each. Okay, I'm gonna replace that out here. So that's going to make my cost function before I are spared. You buy are multiplied by the zero minus 2/3 fire you divided by by our spirit. If we find that a little bit, um, get rid of the I and we can get rid of one of these ours and my cost function then becomes four by art, plus to the zero minus 4/3 pi r cubed all divided by r. Then I'm gonna take the derivative. Hey, before I do that actually, I'm going to split this term, so I'm gonna make it four point arts. What? U V zero over in mine? It's 4 30 Fire squared. Okay. And what that's going to allow me to do is I think what you think. Four minus 4/3. Four is 12 3rd Or he's gonna get the third. My art's weird Plus do V zero over, but hopefully follow that so far. Now, in order to minimize this derivative, C prime is going to equal teen over three pi R minus two. These your over art Weird derivative. I'm now gonna start. You can see that. I'm gonna take that and Senate equal to zero, which means that 16 pi over three are minus. Sorry. It's gonna equal to the zero over our squares. Okay? And what that means is that our cubed if I bring the r squared up is going to equal to V zero, divided by 16 pie divided by three divide boats. I What that's going to do is give me are cube is to the zero a times 3/16 I And if you reduced that, you'll notice this reused eight and then I just take the cube root. So the are is the cube root of three V 0/8 pi. Now that's that. Our dimension, but then fastest for the dimensions that we have to also find each okay is equal to If we go back up here This a v zero minus 2/3 pi r cubed or pi r squared so minus and then scroll back up A 2/3 pi r cubed and then divided by pi Arts cleared if you look back up here so I don't really want all that simplifying. Quite so I'm just gonna divide by Pi r Square, which he's gonna give me this and then it becomes easy. Roll over and a human room. Oh, three V 0/8 pi. I'll squared times by and then minus 2/3. Okay, Now, this is going to get a little funny, but I think you can follow. This is V zero. Hopefully, you can, um, over pi times three v zero, all to the 2/3 they divided by eight. Hi. The 2/3. All right. And then minus 2/3. No. Uh, hi. And plight of 2/3 is gonna leave me with V 0/3. The zero time too high to the 1/3. Okay. And then divided by eight to the 2/3 is actually just four. Okay. And then this three v zero is still to the 2/3. So I'm gonna raise this to the 2/3. You know, you can't see that very well. But then that's going to give me these year old about my visa is here 2/3 its V zero with 1/3 this four is gonna flip up. And then we have divided by three to the 2/3 which is nine. The 1/3 times pi the wondered and then minus 2/3. So what I'm gonna do is I'm gonna take the cube root part, and I'm gonna be left with four cubed, which is 64 I, the zero divided by nine pie and then minus 2/3. And then if I make that into a cube root, I can cube these and make it. And I forgot something Way up here. I miss copied this. This was supposed to be two pi r cubed, which means this is supposed to have an r in it. This is supposed to have a higher in it. And so what I'm gonna do is I'm gonna actually replace are here. I'm gonna supposed to have this cube root of three V 0/8 pi. Okay, This is also supposed to be the cube root of three V 0/8 pi. And this is supposed to be the cube root of three V 0/8. I Okay, so then I get this here. So then we're gonna do a little bit more rearranging here. I'm gonna take this 88 to become just 1/3 to root of BV zero over I in this area. No, it's for you, Root. Um V zero over. Excuse me. Nine pie. So what I want to do is I'm actually gonna take this out, and I'm gonna put a three in here so that if I brought this back in, I get three over to meet seven. Okay? Now, these air like firm. So what is happening is 4/3 minus. The Munford gives me just one of these. There was a lot of algebra, but that's my height.

Okay, We're gonna be starting with a, um, right circular cylinder. So we have a right, um, circular cylinder, and it is topped by 1/2 of a, um, sphere. So it's taught by a hemisphere, Um, and this is actually a silo. So all silo is constructed by a I'm right. Circulars. Um, cylinder topped by a hemisphere. Um, so here is the radius of the hemisphere, which is also gonna be the radius of that circular cylinder. And we have the height of the cylinder and the cost of construction. Um, per square unit of the surface area is twice as great for the hemisphere, which makes sense because of the circular nature of that hemisphere. And we also know that the volume is fixed. So it's a fixed volume, and we want the cost of construction to be minimized. Okay. And so we do know that volume if we're gonna right the volume formula for these, the total volume is, um, the volume of that cylinder which is going to be pi r squared H plus half of a volume of a sphere which is 4/3 pi. Oh, are que and so this is gonna be pi r squared eight plus 2/3 pi r cute. Okay. And so we know costs of construction is going to be based off of the surface area. So let's come up with a surface area total surface area formula, so surface area, the total surface area is going to be and we only have to worry about the lateral area of that um, lado area of the cylinder. And so the lateral area of the cylinder is h times two pi are So that's gonna be the lateral area of the cylinder. Plus, Now we want half of the surface area of the him the sphere, because it's a hemisphere, and so that is only going to be too pie r squared. But we also know that it cost twice as much. So we need to multiply that by two. Okay. And so we want to minimize. So we know we're gonna have to take the derivative of the surface area. And so what we want to do is to right the surface area in terms of only one variable. And so we're gonna use this volume formula to solve for H because we know volume is fixed and So what we have here is that V minus 2/3 pi r cubed, divided by pi r squared is equal to H And so h h is gonna be equal to the over pi r squared minus kind of simplify this to third or and so over here in my surface area formula we're gonna now have two pi r times This volume over pi r squared mind is 2/3 are plus four pi r squared. And so now we're going to do some simplification. Try to get this in a better forms for us to take our derivative and so were you. Simplify this and we get this equal to To the over are minus four pi r squared over three plus four pi r squared And so let's go ahead and simplify a little bit more. And so let's go ahead and pull out, um r squared. And so we'll have an r squared here and where when we simplify that, that is going to be a pie. Over three kids will have a 12 pi minus a four pi Um and then this will be plus to the over our and so that's gonna be a whole lot easier to take the driven above. And so we take the derivative of the surface area with respect to our, which gives us to our times eight pi over three minus to V over r squared. And so let's go ahead and put that over a common denominator and we get 16 pi r cubed minus two V minus 60 so minus 60 because we're putting it over a common denominator of three r squared so 60 over three r squared. And so we know our cannot equal zero. And so now we want to set that derivative. We want to set that derivative equal to zero. Let's get back up there. Let's set that derivative equal to zero. So and so that's only gonna affect the numerator. So 16 pi r cubed minus six b. And so when we do that, we get our is equal to, um three B over eight. Hi to the 1/3. Okay, so that is what we think the radius is. And so let's see, um, if that will be a maximum or a minimum, um, place where my surface area will occur. So let's go ahead and take the um, second derivative. And so when we take thes second derivative, we get, um 16 16 pi over three plus four v over R cubed and we know are, of course has been greater than zero. So no matter what, this will be positive. Which means that this will be a place. Um, where surface area, um will be maximize or minimized. Sorry, cause it's positive. Which means it's Kong cave. Uh um And so now let's go ahead and find r h. So So we know that are is the three v the Cube ooh of three V over eight pi. We also know that h was also equal to, um V over see the minus the 2/3 pi times the, um, our cube. So this would be three V over eight pi divided by, um pie times are squared, and that's gonna be three V over eight pi to the 2/3 power. And so now we have this equal to the minus V over four, divided by that pie. Times three V over eight pi 2/3 power. So this is gonna be equal. Teoh three over four B and then we're gonna multiply both the top and bottom. We're gonna rationalize that denominator. So I have three V over eight pi to the 1/3 divided by the pie times the three V over eight pie. So H now becomes, um if I simplify three v over pie to the 1/3 and our is three V over eight pi to the 1/3. So they're my dimensions that will minimize the cost of the construction.

So first, let's try to find the volume of this cylinder portion, and we'll find along by using. Volume equals pi radius Square times height. The radius of this still central cylinder is half of the diameter of the gamers, 20 than half of it is 10. So 10 square, the high is 60. And if I multiply that altogether, I get 6000 kind. Now let's find the volume of the hemisphere at the top. You can take half of the volume of the entire sphere for thirds pi radius cubed. The radius of the top is the same radius of the cylinder, which is 10. So 4/3 pi 10 cubed. All of that simplifies to 2000 over three tank. So if I want to find the total volume that I would take both of my volumes and add them together on the album are the next page. So 6000 high plus 2000 over three pie. I could make this into a fraction over three by multiplying this times 37 9000 pry Oh my. I have that altogether or not from not 27,000 high when I add that all together I get 20,000 over three type all that simplifies to about 20,943 0.2 95.


Similar Solved Questions

5 answers
The weights of newborn baby boys born at a local hospital are believed to have normal distribution with mean weight of 3728 grams and variance of 432,964_ If a newborn baby boy born at the local hospital is randomly selected, find the probability that the wcieht will be between 4122 and 4583 grams Round vour answer to four declma places.
The weights of newborn baby boys born at a local hospital are believed to have normal distribution with mean weight of 3728 grams and variance of 432,964_ If a newborn baby boy born at the local hospital is randomly selected, find the probability that the wcieht will be between 4122 and 4583 grams R...
5 answers
Problem25 pdlrts)sole U tna sclid oxtalted by rolatrg In region boLnclod becl Lncomouled terg Iha Mithod U wustansa an KilevaMan Itu IKMEAvelian # Ol Ifva bolal€m #c0 CNalcnlartanlhallequaqnico ura _
Problem 25 pdlrts) sole U tna sclid oxtalted by rolatrg In region boLnclod b ecl Ln comouled terg Iha Mithod U wustansa an Kileva Man Itu I KMEA velian # Ol Ifva bolal€m #c0 CNal cnlartanlhalle quaqnico u ra _...
5 answers
Marks) Find the volume of the solid that lies within the sphere 1? +y? + 22 = 4 abore the Ty-plane, and between the cones V3Vr? + y? and < =
marks) Find the volume of the solid that lies within the sphere 1? +y? + 22 = 4 abore the Ty-plane, and between the cones V3Vr? + y? and < =...
4 answers
Select the correct statement regarding the first stable product formed in Hatch and Slack pathway in $C_{4}$ plants.(a) Oxaloacetate is formed by carboxylation of phosphoenol pyruvate (PEP) in the bundle sheath cells,(b) Oxaloacetate is formed by carboxylation of phosphoenol pyruvate (PEP) in the mesophyll cells.(c) Phosphoglyceric acid is formed in the mesophyll cells.(d) Phosphoglyceric acid is formed in the bundle sheath cells.
Select the correct statement regarding the first stable product formed in Hatch and Slack pathway in $C_{4}$ plants. (a) Oxaloacetate is formed by carboxylation of phosphoenol pyruvate (PEP) in the bundle sheath cells, (b) Oxaloacetate is formed by carboxylation of phosphoenol pyruvate (PEP) in the ...
5 answers
44.6 What is the Larmor frequency of protons in hydrogen atomswhen subject to the following external magnetic fields:(a) 0.10 mT (b) 0.15 T (c) 0.95 T (d) 1.5 T (e) 5.0 T(f) 15T
44.6 What is the Larmor frequency of protons in hydrogen atoms when subject to the following external magnetic fields: (a) 0.10 mT (b) 0.15 T (c) 0.95 T (d) 1.5 T (e) 5.0 T (f) 15T...
5 answers
For the following exercises, determine the equation of the parabola using the information given.Focus $(-3,5)$ and directrix $y=1$
For the following exercises, determine the equation of the parabola using the information given. Focus $(-3,5)$ and directrix $y=1$...
5 answers
0-/3 Points]DETAILSGetpil 4.7.p.059,Mi; C/1OD Sut misslons Uscd HY NOTES 4Ma 4t, Sshlemiayd In Eho @uurn teljn Tuvr n Fratloniee AeeAsk Your TEACHERLlockaRACTICE ARotucRJecta Ennan3,5-ko hlckDolerinna Ire ecreeradcng7 rsht}arg0c7nconnettng {ne 3.5-*9 &"0 thc 0-tqueoEreriu1,0 "9dcdk m t ? 0-6obokNeed Help?tettan{Dou" T
0-/3 Points] DETAILS Getpil 4.7.p.059,Mi; C/1OD Sut misslons Uscd HY NOTES 4Ma 4t, Sshlemiayd In Eho @uurn teljn Tuvr n Fratloniee Aee Ask Your TEACHER Llocka RACTICE ARotucR Jecta Ennan 3,5-ko hlck Dolerinna Ire ecreeradcng7 rsht} arg 0c7n connettng {ne 3.5-*9 &"0 thc 0-tqueo Ereriu 1,0 &q...
5 answers
Find the following limitsCO3 TI) lim 41lim , R(z) , (-9z, 2 < -4 where R(z) = (1 - 32, 2 > -1
Find the following limits CO3 TI) lim 41 lim , R(z) , (-9z, 2 < -4 where R(z) = (1 - 32, 2 > -1...
5 answers
VenFy that Y=€-X t 2Xe (s 1 s6luhion T ," + 2y'+y on 4h e in terval C-11 M)
VenFy that Y=€-X t 2Xe (s 1 s6luhion T ," + 2y'+y on 4h e in terval C-11 M)...
5 answers
Evaluate the indefinite integral by using the given substitution to reduce the integral to standard form_1Osec 2 (Sx)tan (Sx)dx,U= tan (Sx) ,U=sec (Sx)Using u = tan (Sx) , 1Osec 2 (Sx)tan (Sx)dx :
Evaluate the indefinite integral by using the given substitution to reduce the integral to standard form_ 1Osec 2 (Sx)tan (Sx)dx, U= tan (Sx) , U=sec (Sx) Using u = tan (Sx) , 1Osec 2 (Sx)tan (Sx)dx :...
5 answers
FanFind the magnitude of the velocity vector of each of the following points, as viewed by a person at rest on the ground: () the highest point on the hoop; the lowest point on the hoop, (iii) a point on the right side of the hoop, midway between the top and the bottom: Express your answers in meters per second separated by commas:Uj, Uji , Ujii5.50,0,3.89 m/sSubmitPrevious AnswersCorrect5.50m S, Uji Om S, Uiii 3.89m
Fan Find the magnitude of the velocity vector of each of the following points, as viewed by a person at rest on the ground: () the highest point on the hoop; the lowest point on the hoop, (iii) a point on the right side of the hoop, midway between the top and the bottom: Express your answers in mete...
5 answers
El_k Iigvdl 4 Uwace kgk 4 gd_fglu tw_al tMM O _ Xk 5ih Alhim_aMd YazoUx) Gsk)s
El_k Iigvdl 4 Uwace kgk 4 gd_fglu tw_al tMM O _ Xk 5ih Alhim_aMd Yazo Ux) Gsk)s...
5 answers
Evaluate the definite integral. f (-+3) (2+3r - 6)" drNeed Help? RaadIlkted Tutor~/1 POINTSTANAPCALC9 6.5.014.Evaluate the definite integral. Hint: let u =x+ 1. 6xvx + 1 dxNeed Help?Read IuTalktol Iutor~[1 POINTSTANAPCALC9 6.5.015, Evaluate the definite Integral: 2xe+2
Evaluate the definite integral. f (-+3) (2+3r - 6)" dr Need Help? Raad Ilkted Tutor ~/1 POINTS TANAPCALC9 6.5.014. Evaluate the definite integral. Hint: let u =x+ 1. 6xvx + 1 dx Need Help? Read Iu Talktol Iutor ~[1 POINTS TANAPCALC9 6.5.015, Evaluate the definite Integral: 2xe+2...
5 answers
The height of a ball at time (in seconds) is given by h(t) =t-92. When is the ball at it's highest point and how high is the ball at picture t0 illustrate this time? Draw YOursolution,
The height of a ball at time (in seconds) is given by h(t) =t-92. When is the ball at it's highest point and how high is the ball at picture t0 illustrate this time? Draw YOursolution,...

-- 0.023507--