Question

If the radius of a sphere is measured as $9m$ with an error of $0.03m$, then find the approximate error in calculating in surface area.

Answer 1

Let $r$ be the radius of the sphere and $\Delta r$ be the error in measuring the radius.
Then, $r=9$ m and $\Delta r=0.03$ m
Now, the surface area of the sphere $\left(S\right)$ is given by,
$S=4\pi r^2$
$\Rightarrow \frac{dS}{dr}=8\pi r^2$
$\therefore dS=\left(\frac{dS}{dr}\right)\Delta r$
$=\left(8\pi r^2\right)\Delta r$
$=8\pi \left(9{)}^2\right(0.03)$ $m^2$ $=2.16\pi$ $m^2$
Hence, the approximate error in calculating the surface area is $2.16\pi$ $m^2$