Class 12 Applications Of Derivatives · Very Long Answer5 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #74

A cone has height h>0 and semi-vertical angle α∈(0,π/2). Show that the greatest-volume coaxial cylinder resting on its base has volume 4πh³tan²α/27.

✅ Step-by-Step Solution

Final Answer: Vmax=4πh³tan²α/27.

Let cylinder height be H and radius R. Similar triangles give R=(h−H)tanα, with 0≤H≤h.
V(H)=πtan²α·H(h−H)².
V′=πtan²α(h−H)(h−3H), whose interior zero is H=h/3.
Its signs are positive before h/3 and negative afterwards, and endpoint volumes vanish, establishing the global maximum.
At H=h/3, R=2h tanα/3, so Vmax=4πh³tan²α/27.

[1 mark] Let cylinder height be H and radius R. Similar triangles give R=(h−H)tanα, with 0≤H≤h.
[1 mark] V(H)=πtan²α·H(h−H)².
[1 mark] V′=πtan²α(h−H)(h−3H), whose interior zero is H=h/3.
[1 mark] Its signs are positive before h/3 and negative afterwards, and endpoint volumes vanish, establishing the global maximum.
[1 mark] At H=h/3, R=2h tanα/3, so Vmax=4πh³tan²α/27.

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