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CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q384 Marksoptimization

CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Question 38 (Applications Of Derivatives)

An open cylindrical tumbler with base radius r, height h and volume V is made with fixed surface area S.
Schematic open cylinder with base radius r and height h.
Schematic open cylinder with base radius r and height h.
Part (1) 2 Marks
Find dV/dr with S held constant.
Answer: (S−3πr²)/2.
  1. Write S=πr²+2πrh.
  2. Solve for h and substitute in V=πr²h to obtain V=(Sr−πr³)/2.
  3. Differentiate with S fixed.
Part (2) 2 Marks
Relate h and r when the volume is maximal.
Answer: h=r.
  1. Set V′(r)=0 to get S=3πr².
  2. Since V″(r)=−3πr<0 for r>0, this interior critical point gives the maximum.
  3. Substitute S=3πr² in S=πr²+2πrh to obtain h=r.

✅ Step-by-Step Solution

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All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/5/1

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