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The volume of a circular cylinder is calculated using theformula V=Pir^2h. If the height h is known to be equal to thediameter 2r and h is measured at the percentag...

Question

The volume of a circular cylinder is calculated using theformula V=Pir^2h. If the height h is known to be equal to thediameter 2r and h is measured at the percentage error of 2%, findthe percentage error in calculating the volume of the cylinder.

The volume of a circular cylinder is calculated using the formula V=Pir^2h. If the height h is known to be equal to the diameter 2r and h is measured at the percentage error of 2%, find the percentage error in calculating the volume of the cylinder.



Answers

The height and radius of a right circular cylinder are equal, so the cylinder's volume is $V=\pi h^{3} .$ The volume is to be calculated with an error of no more than $1 \%$ of the true value. Find approximately the greatest error that can be tolerated in the measurement of $h,$ expressed as a percentage of $h$

Were given a geometric figure with possible errors in the measurements of its dimensions, were asked to approximate the maximum possible error in the measurement of volume. Of this figure, figure is a right circular cylinder, the Radius R and the height of H. We're told that the Radius R is measured with a possible error of 4%. This means that the absolute value of D. R over R is equal 4% and the height age is measured with possible error of 5% which means that the absolute value of D. H over H is equal to 5%. You want to find the maximum possible error and measuring the volume. So first recall that the volume of right circular cylinder is given by volumes. He is equal to area the base pi r squared times the height h Therefore we have that v R r H is equal to two pi r h and V H R H is equal to pi r squared and therefore we have the differential TV. Total differential is equal to two pi r h er plus pi r squared th so we have that the absolute value of the differential of the total differential be over V. This is the possible percent error in the volume is equal to the EPS. The value of two pi r age divided by high R squared h which gives us to e. R over our plus pyre square to D H over pyre squared age which is D H over age. And we have by the triangle inequality that this is less than or equal to two times er over our waas d h over h and we're given the possible errors are 4% and 5% So this is equal to two times 4% plus 5% which is equal to 13%. And so you've shown that an upper bound and indeed the maximum value for the possible percent error in volume is 13.

Problem says the height and radius of a right circular cylinder are equal to the cylinders. Volume is V equals pi H Q. The volume is to be calculated with an air of no more than 1% of the true value. Find approximately the greatest error that could be tolerated in the measurement of H expressed as a percentage of H. So what we do now is we take a differential of TV getting three pi squared the eggs because this is our error and we're saying that Devi has to be less than or equal to 0.1 or 1% of the true value of me or wine. So what we do is we replace everything into this equation right here because we're trying to solve. So we know Devi three high H squared T H less than or equal to 0.1 and then we put in our volume. Hi h cute. What we do now is we solved for D h. So we divide both sides by three pie h squared. So we get th by self all right, right here th in, less than or equal to 0.1 thie H squared cancels out and leaves one h on top. Hi, H or actually that pie also canceled out. So we just have 0.1 h over three. And so we know it's 0.1 over three times a tch is the max that are, ah, error in height can be or H So if it's 0.1 over three and we want to express as a percentage, we multiply this by 100 and we get won over three percent.

Section 3.9 Problem number 47 we're dealing with tolerance here, So they're giving us a right circular cylinder. Um and it says the height and the radius are equal. So normally we know that volume is equal to pi r squared h Okay, if our is equal to H than this formula just becomes volume equal, high, and we've got our choice. But they wanted in terms of age. So it's gonna be either pi r cubed or pry each cube. So we've got the volume is equal to Pi H Cube. Now what they want us to do is I want to say that we want to calculate this volume with an air of no more than 1% of the true value. Okay, So in order to do that, less estimate Devi is equal to So Devi is going to be vey prime of h times D h okay. And that could be my estimate for the differential and then the it says that the volume is no more than 1% of the true value. So this divided by the true value has to be roughly 1%. Okay, so the true value we know is going to be right here. Pie are skewed. Uh, hi. Are high h Q b. Okay, so that is the true value. So what I'm saying is that this estimate So, Devi that estimate that I see there, um, over the actual ve of h um, that's roughly 1% is what I'm after. Okay, So if I look at this, let's figure out then what we would have So the prime of H i v prime of h D H over V of h im saying that that has to equal one. Okay, so I'm gonna have what if I take the derivative here So the prime is going to be, what? Three pie h squared. So I'm gonna have three pie h squared D h over the volume, which was what? Hi h cubed. That has to equal one. And then, in order to simplify this, the pie simplifies this h squared goes away, and I'm just left with h there. So what that tells me is that what three over h d h is equal to one so three over h d h. Because I said it was 1%. That's what I get there. So d h is equal to 1/3 h. So it tells me as a percentage of eight I'm dealing with 1/3 percent or 0.33% would tell me the, um, the tolerated measure of H. Okay, so that's how my tolerance would be for the measure of H to keep that tolerance off. Ah, 1% in the overall volume calculation.

In the question. We have to estimate people. Are you off this, Linda, About how acutely name B I spread it, The calculated from the measurement off and it that are in the other off one person. So now moving towards the solution. One name of this is that it's not goodbye food by audit. And the Dainius is still protected by by artists. Where or the area or is so Davey? They went toe dp Blessed be h b. Yet there s Davey. Bye bye. DP less by square the edge on I already spread at one of the No Diaz by they would be less than that one. And the edge 200 then So the percentage Jeter, come on. We all 3% and this will live on the land for the question. Thank you.


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