MHT CET · Maths · Application of Derivatives

The maximum volume of a right circular cylinder if the sum of its radius and height is
\(6 \mathrm{~m}\) is

A \(16 \pi m^{3}\)
B \(32 \pi m^{3}\)
C \(4 \pi m^{3}\)
D \(64 \pi m^{3}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(32 \pi m^{3}\)

Step-by-step Solution

Detailed explanation

(A)
Let height of cylinder be \(\mathrm{h}\) and radius be \(\mathrm{r} \Rightarrow \mathrm{r}+\mathrm{h}=6 \Rightarrow \mathrm{h}=6-\mathrm{r}\)
Volume of cylinder \(\mathrm{V}=\pi \mathrm{r}^{2} \mathrm{~h}\)
\(=\pi \cdot r^{2} \cdot(6-r)=\pi\left(6 r^{2}-r^{3}\right)\)
For the maximum volume, \(\frac{\mathrm{dV}}{\mathrm{dr}}=0\)
\(\pi\left(12 r-3 r^{2}\right)=0 \Rightarrow 12 r=3 r^{2} \Rightarrow r=4 \Rightarrow\) \(h=6-4=2\).
\(\therefore\) Volume of cylinder \(=\pi \times 4^{2} \times 2=32 \pi \cdot \mathrm{m}^{3}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The maximum volume of a right circular cylinder if the sum of its radius and height is\(6 \mathrm{~m}\) is

Step-by-step Solution

The maximum volume of a right circular cylinder if the sum of its radius and height is
\(6 \mathrm{~m}\) is