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Chapter 1: Problem 95

If \(\mathrm{A}=\mathrm{b}^{4}\) the fractional error in \(\mathrm{A}\) is $\ldots \ldots \ldots \ldots$ (a) \(\left[(\Delta \mathrm{b})^{4} /(\mathrm{b})\right]\) (b) \([\Delta \mathrm{b} / \mathrm{b}]\) (c) \(4[(\Delta \mathrm{b}) /(\mathrm{b})]\) (d) \((\Delta \mathrm{b})^{4}\)

Short Answer

Expert verified
The short answer is: (c) \(4\left[\frac{\Delta\mathrm{b}}{\mathrm{b}}\right]\)

Step by step solution

01

Calculate the absolute error in A

First, we need to find the absolute error in A by differentiating A with respect to b. When we differentiate A with respect to b, we will obtain the factor by which the absolute error in b affects the absolute error in A. \( \frac{\mathrm{dA}}{\mathrm{db}} = \frac{\mathrm{d(b)^4}}{\mathrm{db}}\) Using the power rule: \( \frac{\mathrm{dA}}{\mathrm{db}} = 4b^3\)
02

Calculate the relative error in A

Now, we need to find the relative error in A by dividing the absolute error in A by A itself: \( \frac{\Delta\mathrm{A}}{\mathrm{A}} = \frac{4b^3 \Delta\mathrm{b}}{b^4} \)
03

Simplify the expression

Now, we can simplify the expression by cancelling out b^3 from the numerator and denominator: \( \frac{\Delta\mathrm{A}}{\mathrm{A}} = \frac{4\Delta\mathrm{b}}{b} \)
04

Select the correct option

Comparing our simplified expression for the fractional error in A with the given options, we can see that the correct answer is: (c) \(4\left[\frac{\Delta\mathrm{b}}{\mathrm{b}}\right]\)

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Most popular questions from this chapter

Kinetic energy \(\mathrm{K}\) and linear momentum \(\mathrm{P}\) are related as \(\mathrm{K}=\left(\mathrm{P}^{2} / 2 \mathrm{~m}\right) .\) What is the equation of the relative error \(\Delta \mathrm{k} / \mathrm{k}\) in measurement of the \(\mathrm{K} ?\) (mass in constant) (a) \((\mathrm{P} / \Delta \mathrm{P})\) (b) \(2(\Delta \mathrm{P} / \mathrm{P})\) (c) \((\mathrm{P} / 2 \Delta \mathrm{P})\) (d) \(4(\Delta \mathrm{P} / \mathrm{P})\)

unit of universal gravitational constant is ............ (a) \(\mathrm{kg} \mathrm{m} \mathrm{sec}^{-1}\) (b) \(\mathrm{N} \mathrm{m}^{-1} \mathrm{sec}\) (c) \(\mathrm{N} \mathrm{m}^{2} \mathrm{~kg}^{-2}\) (d) \(\mathrm{N} \mathrm{m} \mathrm{kg}^{-1}\)

Heat produced in a current carrying conducting wire is \(\mathrm{H}=\mathrm{I}^{2} \mathrm{Rt}\) it percentage error in $\mathrm{I}, \mathrm{R}\( and \)\mathrm{t}\( is \)2 \%, 4 \%\( and \)2 \%$ respectively then total percentage error in measurement of heat energy \(\ldots \ldots \ldots\) (a) \(8 \%\) (b) \(15 \%\) (c) \(5 \%\) (d) \(10 \%\)

For a sphere having volume is given by \(\mathrm{V}=(4 / 3) \pi \mathrm{r}^{3}\) What is the equation of the relative error \((\Delta \mathrm{V} / \mathrm{V})\) in measurement of the volume \(\mathrm{V}\) ? (a) \(3(\Delta \mathrm{r} / \mathrm{r})\) (b) \(4(\Delta \mathrm{r} / \mathrm{r})\) (c) \((4 / 3)(\Delta \mathrm{r} / \mathrm{r})\) (d) \((1 / 3)(\Delta \mathrm{r} / \mathrm{r})\)

Dimensional formula of latent heat is ........ (a) \(\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\) (d) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-1}\)
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