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NCERT Class 12 · Ch 6 Exercise 6.3 · Q18

NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Exercise 6.3 Question 18

Cut equal corner squares from a 45 cm × 24 cm tin sheet and fold an open box. Find the cut size maximizing volume.
Optimal box net: cut 5 cm squares
Optimal box net: cut 5 cm squares

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: Cut side 5 cm; maximum volume 2450 cm³.
  1. If cut side is x, the box has base (45−2x)×(24−2x), height x, and 0<x<12.0.
  2. V(x)=\(x \left(24 - 2 x\right) \left(45 - 2 x\right)\); V′=\(12 \left(x - 18\right) \left(x - 5\right)\).
  3. The only stationary point in the feasible interval is x=5. The derivative changes from positive to negative there, while volume tends to zero at the feasible endpoints.
  4. Therefore the maximum volume is V(5)=2450 cm³.

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