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Q10.

Expert-verifiedFound in: Page 18

Book edition
7th

Author(s)
Douglas C. Giancoli

Pages
978 pages

ISBN
978-0321625922

**What, roughly, is the percent uncertainty in the volume of a spherical beach ball of radius $\mathbf{r}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{84}\mathbf{\pm}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{m}$****?**

The correct answer of percent uncertainty is 10%.

**Volume is defined as the amount of space that a particular object or body occupies in the space.**

** **

The radius of the spherical beach ball, $r=0.84\pm 0.04\mathrm{m}$.

The formula of the volume of a sphere is

$V=\frac{4}{3}\pi {r}^{3}\dots \left(i\right)$

From equation (i), the volume is

**$V=\frac{4}{3}\pi {\left(0.84\right)}^{3}$.**

Thus, the final value of the multiplication is 0.6%.

Differentiating equation (i) w.r.t to radius,

**$\frac{\mathrm{d}V}{\mathrm{d}r}=\frac{4}{3}\pi 3{r}^{2}\mathrm{d}r$.**

The formula of the small change in the volume of a sphere is

role="math" localid="1643623569468" $\mathrm{d}V=4\pi {r}^{2}\mathrm{d}r\dots \left(ii\right)$.

From equation (ii), the small change in the volume is

$\mathrm{d}V=4\pi {\left(0.84\right)}^{2}\left(0.04\right)$.

The formula of percent uncertainty is

$V\text{'}=\frac{\mathrm{d}V}{V}\times 100\%\dots \left(iii\right)$ .

From equation (iii), the value of percent uncertainty is

$\begin{array}{c}V\text{'}=\frac{4\pi {\left(0.84\right)}^{2}\left(0.04\right)}{\frac{4}{3}\mathrm{\pi}{\left(0.84\right)}^{3}}\times 100\%\\ \approx 10\%\end{array}$

Thus, the final value is 10%.

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