CBSE 2023 · Region 2 · Set 2 · Q37 · 4 marks
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 cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $\displaystyle 75 \pi \mathrm{~cm}^{2}$. Based on the above information, answer the following questions :(i)If the radius of cylinder is r cm and height is h cm , then write the volume $\displaystyle V$ of cylinder in terms of radius $\displaystyle \mathrm{r}$.(ii)Find $\displaystyle \frac{\mathrm{dV}}{\mathrm{dr}}$.(iii)Find the radius of cylinder when its volume is maximum.For maximum volume, $\displaystyle \mathrm{h}>\mathrm{r}$. State true or false and justify.
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 cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $\displaystyle 75 \pi \mathrm{~cm}^{2}$. Based on the above information, answer the following questions :
(i)
If the radius of cylinder is r cm and height is h cm , then write the volume $\displaystyle V$ of cylinder in terms of radius $\displaystyle \mathrm{r}$.
(ii)
Find $\displaystyle \frac{\mathrm{dV}}{\mathrm{dr}}$.
(iii)
Find the radius of cylinder when its volume is maximum.
For maximum volume, $\displaystyle \mathrm{h}>\mathrm{r}$. State true or false and justify.
Marking-scheme solution
(i)
$\displaystyle \pi \mathrm{r}^{2}+2 \pi \mathrm{r} \mathrm{h}=75 \pi \Rightarrow \mathrm{h}=\frac{75-\mathrm{r}^{2}}{2 \mathrm{r}}, \therefore V=\pi \mathrm{r}^{2} \mathrm{h}=\frac{\pi}{2}\left(75 \mathrm{r}-\mathrm{r}^{3}\right)$
(ii)
$\displaystyle \frac{\mathrm{dV}}{\mathrm{dr}}=\frac{\pi}{2}\left(75-3 \mathrm{r}^{2}\right)$
(iii)
$\displaystyle \left.\frac{\mathrm{dV}}{\mathrm{dr}}=0 \Rightarrow \mathrm{r}=5, \frac{\mathrm{~d}^{2} \mathrm{~V}}{\mathrm{dr}^{2}}\right]_{\mathrm{r}=5}=\frac{\pi}{2}(-6 \mathrm{r})<0 \therefore$ 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 $\displaystyle \mathrm{r}=5 \Rightarrow \mathrm{h}=\frac{75-5^{2}}{2(5)}=5 \Rightarrow \mathrm{h}=\mathrm{r}$
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CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.