Question

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

Short answer

The correct answer of percent uncertainty is 10%.

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, $r=0.84\pm 0.04\mathrm{m}$.

Writing the formula of the volume of a sphere

The formula of the volume of a sphere is

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

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 is0.6%.

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

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

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

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

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

Step 7. Calculating the value of percent uncertainty

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

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

Thus, the final value is 10%.