Question

If the radius of a sphere is measured as $7$ m with an error of $0.02$ m, find the approximate error in calculating its volume.

Answer 1

Let $r$ be the radius and $v$ be the volume of the sphere, then

$v=\frac{4}{3}\pi r^3\Rightarrow \frac{dv}{dr}=4\pi r^2$

Let $\Delta r$ be the error in $R$ and the corresponding error in $v$ be $\Delta v$, then

$\Delta V=\frac{dv}{dr}\Delta r=4\pi r^2\Delta r$

If $r$ given that $r=7$ and $\Delta r=0.02$

$\therefore \Delta v=4\pi \times 7^2\times 0.02$

$=3.92\pi$

Hence, approximate error in calculating volume is $3.92\pi m^3$