• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q10.

Expert-verified
Found in: Page 18

### Physics Principles with Applications

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 $\mathbf{r}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{84}\mathbf{±}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{m}$?

The correct answer of percent uncertainty is 10%.

See the step by step solution

## Step 1. Writing the given data

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, $r=0.84±0.04\mathrm{m}$.

## Step 2: Writing the formula of the volume of a sphere

The formula of the volume of a sphere is

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

## Step 3: Calculating the volume of the spherical beach ball

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%.

## Step 4. Writing the formula for a small change in volume

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

## Step 5. Calculating the value of a small change in volume

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

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

## Step 6. Writing the formula of percent uncertainty

The formula of percent uncertainty is

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

## Step 7. Calculating the value of percent uncertainty

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

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

Thus, the final value is 10%.