Mathematics - Application of Derivative Question with Solution | TestHub

$12^2=r^2+{\left(\frac{h}{2}\right)}^2\Rightarrow V=\pi r^{2}h$ $\Rightarrow V=\pi \left(144-\frac{h^2}{4}\right)h$$\Rightarrow V=144\pi h-\frac{\pi }{4}{h}^3\Rightarrow \frac{dV}{dh}=144\pi -\frac{3\pi }{4}{h}^2$ $\Rightarrow \frac{dV}{dh}=0\Rightarrow 144\pi =\frac{3\pi }{4}{h}^2$ $\Rightarrow h^2=48\times 4\Rightarrow h=8\sqrt{3}$ $\therefore 12^2=r^2+48\Rightarrow r^2=96$ Volume $=\pi r^{2}h=\pi \times 96\times 8\sqrt{3}=768\sqrt{3}\pi {\mathrm{cm}}^3$.