Mathematics - Application of Derivative Question with Solution | TestHub

Let $r$ be the radius of the sphere and $\Delta r$ be the error in measuring the radius. Then, $r=9\mathrm{cm}$ and $\Delta r=0.03\mathrm{cm}$ Let $V$ be the volume of the sphere. Then, $\begin{array}{l}\mathrm{V}=\frac{4}{3}\pi r^3\\ \Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=4\pi r^2\\ \Rightarrow {\left(\frac{\mathrm{dV}}{\mathrm{dr}}\right)}_{r=9}=4\pi \times 9^2=324\pi \end{array}$ Let $\Delta \mathrm{V}$ be the error in $\mathrm{V}$ due to error $\Delta \mathrm{r}$ in $r$. Then, $\begin{array}{l}\Delta \mathrm{V}=\frac{\mathrm{dV}}{\mathrm{dr}}\Delta r\\ \Rightarrow \Delta \mathrm{V}=324\pi \times 0.03=9.72\pi {\mathrm{cm}}^3\end{array}$