Question

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

Answer 1

Let $r$ be the radius and $S$ be the surface area of the sphere. Then,

$S=4\pi r^2\Rightarrow \frac{dS}{dr}=8\pi r$

Its is given that $r=9$ and $\Delta r=0.03$. Therefore error $\Delta S$ in $S$ is given by

$\Delta S={\left(\frac{dS}{dr}\right)}_{r=9}\Delta r=8\pi \times 9\times 0.03=2.16\pi m^3$.