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Chapter 1: Problem 101

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) \((1 / 3)(\Delta \mathrm{r} / \mathrm{r})\)

Short Answer

Expert verified
(a) \(3(\frac{\Delta r}{r})\)

Step by step solution

01

The formula for the volume of a sphere is given as: \(V= \frac{4}{3}\pi r^3\). #Step 2: Determine the differential of the volume with respect to r#

We will now differentiate the volume formula with respect to the radius r. This will give us \(\frac{dV}{dr}\). \[\frac{dV}{dr} = 4\pi r^2\] #Step 3: Compute the change in volume ∆V#
02

Now we will use the differential, \(\frac{dV}{dr}\), to find the approximate change in volume, ∆V, for a given change in the radius, ∆r. \[\Delta V \approx \frac{dV}{dr} \Delta r = 4\pi r^2 \Delta r\] #Step 4: Calculate the expression for relative error#

The relative error is the ratio of the change in volume to the original volume, which can be written as \(\frac{\Delta V}{V}\). Using the expressions from step 3 and step 1, calculate the relative error formula. \[\frac{\Delta V}{V} = \frac{4\pi r^2\Delta r}{\frac{4}{3}\pi r^3} = 3\frac{\Delta r}{r}\] Comparing the calculated relative error expression with the given options, we find that the answer is: (a) \(3(\frac{\Delta r}{r})\).

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Most popular questions from this chapter

Dimensional formula for thermal conductivity (k) is.. (a) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2} \mathrm{~K}^{1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{-3} \mathrm{~K}^{-1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-3} \mathrm{~K}^{-1}\)

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