Question

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

Answer 1

Let $r$ be the radius of the sphere and $\Delta r$ be the error in measuring the radius.
We have given, $r=7$ m and $\Delta r=0.02$ m
Now, the volume $V$ of the sphere is given by,
$V=\frac{4}{3}\pi r^2$
$\Rightarrow \frac{dV}{dr}=4\pi r^2$
$\therefore \Delta V\approx dV=\left(\frac{dV}{dr}\right)\Delta r$
$=\left(4\pi r^2\right)\Delta r$
$=4\pi \left(7{)}^2\right(0.02)m^3=3.92\pi$ m${}^3$
Which is approximate error in measuring volume