the-circumference-of-a-sphere-was-measured-to-be-84-mathrmcm-with-a-possible-error-of-05-mathrmcm--2

Question

The circumference of a sphere was measured to be 84 $\mathrm{cm}$
with a possible error of 0.5 $\mathrm{cm} .$
(a) Use differentials to estimate the maximum error in the
calculated surface area. What is the relative error?
(b) Use differentials to estimate the maximum error in the
calculated volume. What is the relative error?

Jack Chen · Accepted Answer

in this question, we are given the circumference of a sphere. Certainly not this circumference by thanks. And from this information, we know the Raiders office a free is given by X over two. Now, we want to talk about the volume and, uh, the theory area for the severe. So part of a issa Paula Boring. So the Valentino by Kendall v equals two. Um, sir, the pasta is actually the, uh, the area. So I did not kept away for the area, which, given by four pi r square and just replaced the radio The radius up in terms of X So we have full pi times x squared, divided by full pi square. So we have X square over pot. It&#x27;s our, um, area And these different show you caused to two x over a pie The X Solan mix era, which is a A, which is to x the X divided by pi and the X and V X is giving in this program and they just Prague in those number. So don&#x27;t forget, um, we have the on the unit Indian, so we used to use the calculator. We have this result 27 squares and Demeter and the full of relative era. We do the question d a over. Hey, the different show divided by ah, the original function. And we need to do some consolation before we plug in the values. So a because the X square over type you know, we have to two times the X divided by X. Now we can just plug in the X and the X. Given my giving, the problem reaches around 0.12 And for property we grow sending that we replace a by the which with the volume. So we for severe because toe 43 pi r cubed and then remember our I closed toe X over two pies. It&#x27;s for three times times X cubed over eight pike you which is X cubed over a six pi square in the end sort of different show you coast to X square over two pi square PXE Not for the Mex era Again we plug in the value off X and V X Even in a program we have 84 square times to your 840.5 divided by two pi square. It&#x27;s around 17 117. I aren&#x27;t your big sentiment. Er so don&#x27;t forget we have this unit and the relative Hera. It&#x27;s given by TV, divided by V. So we again we go some consolation before we probably in the value for X and P X. So we have three d x over x Simeon. And now we can just plug in the values for external yanks. So we have 0.18 and their percentage herro just converse. This result to be a percent 1.8%. And we do the same thing for this, uh, for this area. So for the area, the percentage very close to 1.2%.

Carlos Pinilla · Answer

So you have some. It&#x27;s you well on way. Measure that their radios, he&#x27;s, uh, inches inches for radius some it&#x27;s here or pull that you will. So you want to computer? What is it? Volume. So you all know what&#x27;s troubling, but there is a the the reduces measure up to, uh, cure asio 0.2 inches. So these, uh, possible here his own error. So some error in then in their radios can be seeing us, uh, build the harder on their will volume. It&#x27;s so fortunate. Radios More specifically, it&#x27;s for thirds. Bye. Are cube. So those peace they&#x27;re in there? It used translate todo there, buddy. Um um, also, it can be approximated by me because all these will be your toe. Both volume with respect there readies terms. Yeah, which is, uh, dealt the heart. So, uh oh, the error volume be approximated like this formula. So it&#x27;s computer. What is now? So you have that be prime is gonna be If he comes down, he&#x27;s gonna be out. Bring comes for thirds. Bye. On the power gets our Lord one. Have a good day. Your crime is this, uh so for this particular case, then in the volume can be approximated to be for bye that I&#x27;m stayed squirt Yeah, inches square times, huh? DDR which is Shiro Point 02 I just so eso So he gets, uh, four times in 12 disease 64. Oh, for Bye. So these number will be, uh, these D&#x27;oh six making multiplied, Dad. Bye 0.0 two. So it&#x27;s gonna be all these numbers are gonna be also these harder. It&#x27;s just a squirt. He&#x27;s gonna be a dangerous thing. Just squared so that the units will be just cute. What&#x27;s up? Well, you should be so from here, you have that two of these. That&#x27;s good. Yeah. D&#x27;oh One What? They should be goto Bye. 12 then thumbs. So it should be the error. Should be about bye terms. Why, um five point so cute on the wall. Similarly, the same court is there for Syria. He&#x27;s sir, Is there you? Depends on the radius is equal to 4/3 support for by hard R squared. So these, uh, error in the radios it&#x27;s gonna be give up on terror. The surface area which can be approximated right differential surface area. So these happens. People too for a video. There you go. Far this surface area. We respect our It&#x27;s gonna be ableto yes, writing times are it is equal to seems the r Does he look to Delta are Well, we go to ISS Brian. Damn So buildup aren&#x27;t so well or as being will do that its prime vehicle to So you two come sounds too them&#x27;s or bye on the power on our gets lowered them&#x27;s for But are that&#x27;s you are I&#x27;ll see how, but they should be. So he right here this is eight aid it That&#x27;s why I&#x27;m saying are 80 square then that that times point or two again? Just cube. So this will be equal to Mmm Oh, he&#x27;s too. That is 64 about thumbs through. These want eight, but you won&#x27;t deploy my point. Oh, one. So, uh, these errors is about while I ate just It&#x27;s good there then right off the Syria. They&#x27;re well forgot. Device. God damn spy on the air in the service of Mobile Bay volume is gonna do right by them. My point

Amrita Bhasin · Answer

Okay. The driven of six X squared is 12 ax. Therefore, we know the maximum error 12 times, 30 times 0.1 gives us 36 centimeters squared. Therefore, relative errors. The area over the surface area therefore 36 divided by six times 30 squared gives 0.0 6/7 on then this ring as a percentage times 100 would be 0.67% for part B. We know that if us his pyre square that s prime of our In other words, the derivative would be two pi r. Therefore, we have to pie times 24 times changing our gives 9.6 pie centimeters squared.

Jack Chen · Answer

in this programme. We No way. We are given her, um, disco with radius and the room to find out the maximum era for the area. So for this, the area is given by Para Square. So I think not veto lower case out to be the Radius and the captor. I took the area in a different show because to two pi r yeah. So the mix era is just the A And we already know what&#x27;s a radius and the words like PR, which is even in the program. So we can just directly Prague in those values. So a nano have 9 26 pie. Don&#x27;t forget we have a unit and the relative era is given by D A over a and the word in out of the air in a So just probably in those functions in the some consolation and, you know, have twice of PR over in the beginning project beyond Dara. So we have, um, two times 0.2, divided by 24 seats around 0.17 and the present age Kara for this area, we just cameras this previous results to be a a president, remember, which is 1.7%

Robert Leedy · Answer

in order to solve this question. You know the equation for volume of a sphere and the equation for service area sphere. I have provided both of those or you. So aside from that, we know are starting radius is 25 centimeters and wants to know what the estimate maximum error is if our radius is within half a centimeter. All right, so we&#x27;re gonna do this twice. We need to find me prime, which is equal. Thio or I r squared every defined rs a prime which is equal to a high art, of course, but that doesn&#x27;t obtain using our power l a. To find our dumps and b we simply plugging over our most by fire change Another So four times 25 I swear I&#x27;m supporting it. Type this directly into a calculator and we get a delta V are changing volume rocks, family evil too. 3927 centimeters. And I&#x27;ll do the same thing for a service area. So we need to play an alarm off by adults arm. So don&#x27;t s a When people to pigs high times 25 times point Father, that is equal. Thio. Approximately 314.2 centimeters squared

Amrita Bhasin · Answer

We know that if we are vey Alexis X Cube than Devi over DX In other words, the derivative is gonna be three x squared. Therefore, we know three X squared times Delta acts is Delta V. Therefore we know relative errors error over the volume. So to 70 we got this by doing three times 30 squared times 0.1 over 30 cubed 0.1 0.1 The relative air times 100 gives us 1% is the maximum errors to 70 centimeters cubed relative a rose 0.1 and percentage error as a percent would be 1%.

Problem

Use differentials to estimate the amount of paint…

Problem 23 Medium Difficulty

Answer

A. Maximum Error in the calculated surface area is 27 $\mathrm{cm}^{2}$
Maximum Relative Error $\approx 0.012$
B. maximum error: 179 $\mathrm{cm}^{3} \quad$ relative error: 1.8$\%$

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A. Maximum Error in the calculated surface area is 27 $\mathrm{cm}^{2}$Maximum Relative Error $\approx 0.012$B. maximum error: 179 $\mathrm{cm}^{3} \quad$ relative error: 1.8$\%$

Essential Calculus Early Transcendentals

Grace H.

Catherine R.

Heather Z.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Grace H.

Catherine R.

Heather Z.

Michael J.

A. Maximum Error in the calculated surface area is 27 $\mathrm{cm}^{2}$
Maximum Relative Error $\approx 0.012$
B. maximum error: 179 $\mathrm{cm}^{3} \quad$ relative error: 1.8$\%$

James Stewart

Essential Calculus Early Transcendentals