Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The…

$\displaystyle \pi \mathrm{r}^{2}+2 \pi \mathrm{r} \mathrm{h}=75 \pi \Rightarrow \mathrm{h}=\frac{75-\mathrm{r}^{2}}{2 \mathrm{r}}, \therefore V=\pi \mathrm{r}^{2} \mathrm{h}=\frac{\pi}{2}\left(75 \mathrm{r}-\mathrm{r}^{3}\right)$

$\displaystyle \frac{\mathrm{dV}}{\mathrm{dr}}=\frac{\pi}{2}\left(75-3 \mathrm{r}^{2}\right)$

$\displaystyle \left.\frac{\mathrm{dV}}{\mathrm{dr}}=0 \Rightarrow \mathrm{r}=5, \frac{\mathrm{~d}^{2} \mathrm{~V}}{\mathrm{dr}^{2}}\right]_{\mathrm{r}=5}=\frac{\pi}{2}(-6 \mathrm{r})<0 \therefore$ volume is maximum when $\displaystyle \mathrm{r}=5$

Or False,

$$\left.\frac{d V}{d \mathrm{r}}=$\displaystyle 0$ \Rightarrow \mathrm{r}=$\displaystyle 5$, \frac{d^{$\displaystyle 2$} V}{d \mathrm{r}^{$\displaystyle 2$}}\right]_{\mathrm{r}=$\displaystyle 5$}=\frac{\pi}{2}(-$\displaystyle 6$ \mathrm{r})<$\displaystyle 0$ \therefore \text { volume is maximum when } \mathrm{r}=$\displaystyle 5$

$$As volume is maximum at $\displaystyle \mathrm{r}=5 \Rightarrow \mathrm{h}=\frac{75-5^{2}}{2(5)}=5 \Rightarrow \mathrm{h}=\mathrm{r}$