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While measuring the radius olla sphere an error ofl2%was inalres his Wqua Wake a0 lapproximate percent errerlin the computed volumelof...

Question

While measuring the radius olla sphere an error ofl2%was inalres his Wqua Wake a0 lapproximate percent errerlin the computed volumelof

While measuring the radius olla sphere an error ofl2%was inalres his Wqua Wake a0 lapproximate percent errerlin the computed volumelof



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Measurement Error A sphere has a radius of 5.81 in., with a possible error of $\pm 0.003$ in. Estimate the maximum error in the volume of the sphere.

And this problem, we have to use the error of radius that were given to find the maximum error of volume of the sphere so far. So let's take a note of a couple things that we're going to need. It's just notation, equations or our values. So we're going to say Let v b the volume of a sphere R B the Radius and we should know that the volume of a sphere is equivalent to 4/3 pi r squared and we're told that our is 5.81 and Delta are Our error is plus ra minus 0.3 So now let's just simply remind yourselves that the volume of a sphere is for over three pi r squared. So what we can do is we can take the derivative of this function will say that D V is equivalent to four or three pi times three R squared in D. R. Remember, we're differentiating with respect to our so that means Delta B would be roughly equal to 4/3 pi times. Three R squared times Delta are so then we can plug in the values for our adult. Are that we had before to find the maximum error. So we get 4/3 pi times. The radius 5.81 will square that multiplied by plus or minus 0.3 That's the error were given. So then we would find when we simplify this this is plus or minus 2.405 pi, which is roughly saying that this is equivalent toe 1.273 inches cubed. Now what we found was the maximum error that could be taken into account when we take the volume of this fear. So I hope that this shows you and explain to you how we can use not on Lee the equation for the volume of the sphere and it's derivative, but also how we can plug in the radius and the error of the radius to find the maximum error off our volume.

So you have a, uh, follow in shape? No, Uh, hawks here. It has, uh, pretties for Thio more for do three foot. Um, as they could be a little My piece. He's sort of like the shape and, uh, well, gobies yeah. This is the actual sex. I'm not. Why? These are the shape this curve will be square with foursquare just 16 my nose x squared. What is the's over? And that girl they What? Why? And also thes part below is mine. Um, so three d is this stuff will be like What do you have that shape? Imagine that you t so that you, Jill, office for you. I hope you can use your imagination too. See what I mean? They know. Oh, we're going approximated. Bye. Eight intervals. Where? Each of these house? Yes. You know that these he's gonna be a disc off these. Thank you. Are these leg that? So you double works, and maybe in the Syrian, they're all the house here bashing that looks a little bit with that. Some shells on, so Well, I think that Yeah. So, uh, well, you're if you turn that he's gonna look something like tip turned up about 90 degrees. We should look sort of like a shape like these. Um, so there any really, well, just three levels, but yeah, So we're gonna do eight. So, divisions, the divisions, you know, lengthen built up don't eggs equals 0.5 billing for each division when we're gonna start like you like them over estimate of this area. So we think they left in. So, uh, no. Yeah, her only if the work's invention. Yeah, so go for one of these. Oh, these one of the thieves discs. Campesinos like these where you helped they're re use. You have some height. They hide here is gonna be in built bikes. I said, I don't know. The volume of these will be. So this helps us in our area. She's the area about this, Chris. All right, R squared so that by our square times the height he's gonna give you this volume that has a total volume. Bye. All right, square terms. No thanks. So thes. Total 1,000,000,000 can be reading us parting much of those pieces so that hold on there basically years like you can you can see these shaped box made it map looks 1,000,000. This Schapelle I ended his license well is not exact. These are approximations, Bradley. I think the volume off each of those is like That's their rough idea. So, uh, so we're gonna Well, well, that book's made it area will be. Does the pics times, huh? By our square, By being well, which of those is gonna be? Well, good bye bye. So we can accorded by being our square for the 1st 1 They're gonna be 16 for the 2nd 1 It's gonna be, uh, six Tiu my nose 0.5 squared, right? He's the square of that function. So each of these will be the corresponding. Why at the point, White Squirt, next one is gonna be so 16 miners wanna square. Certainly. 15 for the next one is gonna be 13 boy 75 Which the sequel to 16. Minus 1.5 square. Right? Because each of these points he's Well, the first one's your RV zero. The X here goes by zero six of one is gonna be I see your 0.5. The next point will be one. Then they exploit Will be one on Bye. The next one is gonna be too, so that things one should correspond to 2 16 minus two squared Schickel Dome. This is equal to fill. It's extremely square Plus on Damon Makes one. It's gonna be my bye makes b seven. The last one is gonna be plus the board. So any foot. So as you can see here we have hey, terms because we're I want to see War Barn seek seven. Hey, because we were caught in this area into hate his laces so well, his number then taking there You used to be. She won five is the world, Really? It's gonna be a toe by James. Might be we knew, Or until one hymns here a point Bye, which is, uh, these himself being He's not exact, but brooks in Medellin because we approximated by 1 46 point oh, nine very and the real value more so. It's one approximation, but this one is more accurate. You see, the forward 2/3 bye are cute. We are equals four. The approximate value will be 4 to 1 30 four 0.4 So that well that works. You made it not approximated these number like that so we called these one. This one is, uh is when he's, uh really would be. This one is like a piece of eight that eight for the eight slices approximation so that, um oh, gee, bring us be surveyed. Divided by V These number he's gonna be a vote Ciro Point who? Hey, what's your point? 089 So that the one here with correspond to our nearer 100%. So these runkle response to one you are about 9% because of being nine years old. Walton Ah, honey. Roar of over about 8.9%. She's broke with 100%. So what you get is is so spirit quit.

So one estimate the allowable allowable percentage error and measuring the Tai Nam ITER of a sphere. If the volumes to be calculated correctly too, then three percent. Okay, so what is the volume of a sphere? What's four thirds pi are cute, but we're talking about the radius. So let's plug in D over two for the radius as give you four thirds Hi de cute over eight or high de cute over cancel for left of the twos, two times excess. Okay, what was the school? The volume of a sphere in terms of a diameter. And so let's say we want TV. It's fine. Devi it's going to be Hi he squared over to because we'll bring down the three and that i'll cancel right and c So what is the maximum error in the volume To be three percent. That's relative relative change to be three percent. I mean, this is no missing a t there, so this is going to be high d squared d over too all over Pi t cubed over six six. Okay, bye. Describes Cancel the six jumps on top cancels with the two becomes the three. So you get three the D over being and then I can divide by three. And I see that them maximum ah, allowable air and the measurement of the diameter is one person.

The given problem here where the volume of a sphere which we know to be B equals 4/3. Hi, are cute. We want to find volume the change in volume with respect to the change in the radius that's going to be we'll find the three here that gets rid of this and I've just become squared. Then we're going to multiply this D. R differential over here so that's what we're having And then we know that this is going to be that it must be kept within .3%. So we're going to have a .03 right here to represent the percentage. So we see that multiplying that times is that their values? We end up getting that the um the maximum permissible percentage error is going to be Plus or -1%. And that would ensure that it stays within the correct amount of volume.


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