Problem

The radius of a circular disk is given as 24 $\ma…

01:45

Problem 21 Medium Difficulty

The edge of a cube was found to be 30 $\mathrm{cm}$ with a possible error in measurement of 0.1 $\mathrm{cm} .$ Use differentials to
estimate the maximum possible error, relative error, and
percentage error in computing (a) the volume of the cube
and (b) the surface area of the cube.

Answer

A. Maximum Error $=270 \mathrm{cm}^{3}$
Relative Error $=0.01$
Percentage Error $=1 \%$
B. Maximum Error $=36 \mathrm{cm}^{2}$
Relative Error $=0.0067$
Percentage Error $=0.67 \%$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 2

DERIVATIVES

Section 8

Linear Approximations and Differentials

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Video Transcript

we're so in this question, condensed apart for part A. We want to find out of Ivorian. So the volume of a cube. So I used the veto the amount of boating, and I use X to denote the lens off the edge. Um, so the morning because the x cubed and we can see that volume V is a function in terms of X, said a TV. The different show equals 23 x square times DX. So first, the Mex era, um, mix era at X equals toe 30 centimeter and that the XY close to 0.1 centimeter. Um, Damien's by the definition of mix area just means TV equals 23 times X, where times DX and plugging those values. We have three times 30 square times 0.1, which is 200 and 70. Um, don't forget you meet. It's a cubic centimeter. The next thing is the relative. So the root of error is just do you find by get happy over V, which can be approximated by TV over V because we already know what's the TV? Was levy here, so just plugging those functions. So it's three X squared times DX Do you buy that by X square? Execute So can cancel out some terms here. So this is the formula for the relative area in this case. So when the X equals is your 0.1, the X equals 30. We have this we crossed with 0.1 and their percentage era is just 1%. We just, um, comforts the relative Aryan Cure percentage number also for party witless. Anything for the area. So party. So I can noting the area by a But in this case, the area of curious six X square. Don't forget we have a six because of Cuba has ah six faces the differential because took 12 x t x and with less anything Nextera sda. Because 12 extends the excellent Prodi in X and V X into this formula, we have 36 send him in the square are square of cinnamon and that a relative era is given by the A over A, which is 12 x times the X, divided by six X square, so have 12 times the eggs over six x, which is two times the egg soul regs. In the problem values we have this equals two. The opening 0067 It's a wrong to this number into the person page era just can run this preface, result to be a person did so it's 0.67%.

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