solve-the-given-problems-by-finding-the-appropriate-differentialif-the-diameter-equals-the-height-wh

Question

Solve the given problems by finding the appropriate differential.If the diameter equals the height, what is the volume of the plastic that forms a closed cylindrical container for which the radius is
$18.0 \mathrm{cm}$ and the thickness is $2.00 \mathrm{mm} ?$

Taylor Shimono · Accepted Answer

So for this problem, we know that we are given a cylindrical tank with a diameter five meters in the height of 20 meters. And we want to know what the error in our volume would be. Something that we&#x27;re going to need your or volume formula, which is going to be equal to pi r squared H. And in order to find our error that occurs from impart a height of 20.1, we&#x27;re going to take the derivative of this equation for volume. With respect to H. That means that DVD is just going to be equal to pi r squared d h. And from here we know that our radius, which is half of our diameter, is just going to be 2.5 meters. Um, so we play that in 2.5 squared and D H is the difference in our age. Value from are given said 20.1 minus 20 is going to be 0.1. So when we saw our error in our volume here is going to be equal to about 1.963 cubic meters and for part B were essentially doing the same thing. The only difference is we&#x27;re told that our diameter now is going to be five foot one meters in our height will be 20 meters. I&#x27;m so again we&#x27;re going to use the volume formula. So our volume is equal to pi r squared H. But this time since we want to know what the change in the diameter, what would affect the change in the diameter will have That means that we&#x27;re going to you want to take the derivative with respect to our So when we do that, we get Devi is equal to two pi r h d r And so our change in radius is going to be ah half of this change in our diameter So d R is going to be a 0.5 our radius. We know original SUV 2.5 high is 20 a n d. R is 0.5 so I mean we&#x27;ll find this out. Our error in our volume in this case is going to be about 15.708 cubic meters

Ankit Gupta · Answer

we have given that the volume each food here volume is given as four feet cube off a cylinder. Celinda, No, no, we have to construct. They&#x27;re still in it. That requires a minimum amount of material. So for this surface area, should we minimum so here volume equals toe by Are the square multiplied? Irritated? That is four and then EJ equals two full development by the square of the city. Question number one. And now for the so far city, addicted by our edge blessed by schools Because this is the open and now we want this valued as minimum. Now we know the value here it&#x27;s that is, by multiplied with its that is full divided by r squared plus bye bye. Solving we get s equals two four divided with our plus by the script. Well, we have to sketch whisker and find the corresponding will you off are so for here. Here is the graph corresponding toe here s equals to four. Divide with our plus for virus crap. So minimum value is here. See this? Several say one is the minimum value. So now the corresponding when you will be here ceiling and the Good morning. A value is our leader here. One they&#x27;re doing so r equals to one. And no put here at that is full divided with by R squared will give full development by did you want. And now this will be one point 2732 So we have to keep our equals to one and h equals to 1.2732 So these other diamonds is

Sarah Lewites · Answer

So in this problem, were given a cylinder and were given some information about the cylinder. You know, it&#x27;s volume only know what service Veria and our job is to find the dimension. So the radius r and the height h that make the cylinder possible under we&#x27;re indicated that we want to choose the smallest possible radius. So that indicates to us maybe we&#x27;ll have a few different possible values for our We want to take the smallest one in order to find the final dimensions of this Selinger. Okay, so the first thing we need to do here is make a system of equations using the information that&#x27;s given writing it in terms of R and H. So for volume, we know that the formula for volume of a cylinder is pie. R squared each. It&#x27;s like base times height. The base will be pyre squared, the highest H. So we have pyre squared H equals 3 20 pie for surface area. Our formula for service area is two pi r squared each of the circles on top and bottom will be pi r squared an area and we have two of them and then plus two pi R each. If you took the round about portion like the label portion off the cylinder, the top would be the circumference of a circle, which is two pi r and the height would be each. So that&#x27;s where we get to Pirate. That&#x27;s gonna be equivalent to 1 92 pi. Okay, so this is our system of equations that we&#x27;re going to use for this problem. Elimination looks to be a difficult strategy for this problem, because the variables aren&#x27;t really matching up in terms of power. So we&#x27;re going to use substitution here on what we&#x27;ll do here is will solve her h in the first equation. And then whatever we get for each will plug that into the second decision. Okay, So just solve for H. We need to divide by pi when you divide by our square, so the pies will cancel. We&#x27;ll get 3 20 over r squared. And now what we can do is our substitution. So this age is gonna go in for this h. So when we do that, we&#x27;ll get this new equation. Two pi r squared plus too high. All right. 3 20 over. R squared in close 1 92 pi. Let&#x27;s go ahead and multiply these two terms together, the two pi r squared. So here one of the ours will cancel and we will have 6 40 Pie on top and just one are on bottom, and that equals 1 92 It looks like we&#x27;re going to have some sort of quadratic or cubic structure. So let&#x27;s go ahead and subtract the 1 92 prior to the other side. It&#x27;s helpful to have things equal to zero when we saw Now what&#x27;s really stopping us ruling for it Here It&#x27;s the divided by R, so let&#x27;s multiply everything by our. When we do that, we&#x27;ll get two pi r cubed. So we&#x27;re gonna be working with a cubic structure. This will become minus 1 92 pi r, and I&#x27;m rearranging them because it&#x27;s typically easiest to factor. If the powers are in descending order and then this will become 6 40 pie. So there&#x27;s no art. All people stare. All right, we&#x27;re going to do some factoring here. Let&#x27;s take out a two pi from everything as it will make our numbers a little bit more manageable as we move into factoring so when we do that, we have our cubed minus 96 are plus 3 20 So this is simply to factor without the pie. Now, we have a great situation here where we have in our cubed, but we don&#x27;t have in our square and we do have an R and we do have a constant term. So when we see something like this, we have to do a splitting the middle term situation. So we have to split this negative 96 are into two terms that allows us to factor by grouping. So we&#x27;re going to split it into minus 16 are minus 80 arm. And the reason why that&#x27;s helpful is that if we continue with factoring by grouping will eventually be ableto have ar minus for in both, and that will be helpful for us. So when we factor by grouping, that means that we take whatever the 1st 2 terms have in common and factor it out and then do the same for the second, too. So it&#x27;s helpful is the only to do two at a time. So when we take from these first, you will take out an arm we&#x27;re left with R squared minus 16. And here let&#x27;s take out a negative ET and we&#x27;re left with R minus four. Okay, now, usually, when we do this factoring that grouping, we really want both of these two terms to have something in common, like a factor in common that we can take out. They don&#x27;t yet, but what we can do is factor this r squared minus 16 because it&#x27;s a difference of squares, and then they will have something in common. So this becomes we thought they are. We are plus four ar minus for And at this point, we&#x27;re feeling good because we have an ar minus were over here, too. So it looks like we&#x27;re gonna be in business, a lot of pregnancies. So now our next move is to take out an AR minus four from this big time in this picture. So we take out in R minus four, and what we have left is our times are plus four minus 80. All right, so let&#x27;s go ahead and distribute. This are through so that it&#x27;s a quadratic try no meal because we might get a solution from this. We&#x27;re definitely going to get a solution from the armistice for So we&#x27;ve got our minus for I&#x27;m not getting the double parentheses because multiplication is communicative and associative, so it means the same thing. And this is our squared. Plus four R minus 80. Okay, so now it&#x27;s time to get some solutions. Now, from the R minus four factor, we get the solution R equals for so radius of four units. What was the unit here in meters. So four meters. Now, from this time we don&#x27;t, it doesn&#x27;t actually factor. So we&#x27;re gonna have to use the quadratic formula. And you should get out a calculator for thought, not a graphing calculator. Register regular calculator. So we&#x27;ll be doing our equals negative for plus or minus the square root of 16 minus four times. Once a market narrated times a negative ET all over too. And this is gonna be negative for it. Plus or minus the square root of 3 36 We were, too. And so then there&#x27;s two possibilities here because we have the plus or minus. We get our equals 7.165 or are equals negative 11.16 So we&#x27;re going to reject the negative right away because we&#x27;re not gonna have a negative radius. We are also not going to use this one. Well, right. Walk rejecting them. Negative. Because we&#x27;re asking the problem to minimize the radius. We want to pick the smallest possible radio. So we&#x27;re not gonna take this one either, because we want the smallest. So our solution that we will use for our is four r equals four meters. Okay, so now we&#x27;ve got the radius dimension, but we still need to get the height dimension. So we just need to plug in four for our into an equation that has each in it. Let&#x27;s use this one. It&#x27;s already solved for H, so we&#x27;ll just need to dio 3 20/4 squared. All right, so, um, we go h equals 3 20 over four. Screw just 16. And that gets us 20 years. So here is our solution. The radius will be four meters, the height will be 20 meters and that our final dimensions to get us our cylinder

Ankit Gupta · Answer

it is given that the lend off the cylinder is the radius plus 20 feet are is the variable we will use to present the radius. Now the volume is given us 984 cubic. So now the volume of the coons on the two sides to go on so two times wondered by r squared on the height is given a stew close. Now the volume of the cylindrical part will be by all of squid and the height or the length in this case is our plus 20 is equal to 984 by no by being a common factor in these two terms and we canceled out with the dome on the right side. So on my plane they get food. Third Ah, squid close. Our cube plus 20 are square on. Expanding is equal to 900 me for so our q plus Upon adding we get 64 by three. Our square is equal to 984. No free money played me three. We get three r Q plus 64 R squared minus 984 minus 980 for money into three were 2952 is equal to zero. No, we can see here in this equation that the number of science in just is equal to number off positive looks. According to discards theory, off signed changes, and that is one. So we can conclude that there is only one positive route, and that is the only road leading to find. So, taking the coefficients, we will lose synthetic division toe, find the positive room. So we will do with six. Six threes are 18 so 18 plus 64 gives us 82 6 teams 82 is for 92 and six times for 92 is 2952 so we can confirm that six is the positive route. Now we&#x27;re not finding the other routes because we know that there is only one positive look. So the remaining routes are either imaginary. What? Negative. So the radius can only be a positive number. So the radius off the cylinder is the only positive route that we have found. That is six inches

Robert Daugherty · Answer

and 3.9 problem 39. We&#x27;re giving the formula in the volume of a right sermon. Circular cylinder. Is he good to pi r squared h And I&#x27;m looking to have a differential that estimates what happens when the radius goes from our zero two are zero plus d r. So if I differentiate this, I&#x27;m going to get what two pi r h. So my differential estimate Devi is going to be too high that point our sub zero h times d r

Problem

Solve the given problems by finding the appropria…

Problem 31 Easy Difficulty

Solve the given problems by finding the appropriate differential.If the diameter equals the height, what is the volume of the plastic that forms a closed cylindrical container for which the radius is
$18.0 \mathrm{cm}$ and the thickness is $2.00 \mathrm{mm} ?$

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus