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Class 12 Applications Of Derivatives · Very Long Answer5 Markscost_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #63

An open-top rectangular tank has depth 2 m and volume 8 m³. The base costs ₹70/m² and the sides ₹45/m². Find the least construction cost.
Open rectangular tank: plan and section
Open rectangular tank: plan and section

✅ Step-by-Step Solution

Final Answer: ₹1000, at base 2 m×2 m.
  1. Let base length l>0 and breadth b>0. Volume gives 2lb=8, so lb=4 and b=4/l.
  2. The base cost is 70·4=₹280; wall area is 2·2(l+b), so C(l)=280+180(l+4/l).
  3. C′=180(1−4/l²)=0 gives l=2 and b=2.
  4. C′<0 for 0<l<2 and C′>0 for l>2, proving the global minimum.
  5. Cmin=280+180(2+2)=₹1000.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let base length l>0 and breadth b>0. Volume gives 2lb=8, so lb=4 and b=4/l.
  • [1 mark] The base cost is 70·4=₹280; wall area is 2·2(l+b), so C(l)=280+180(l+4/l).
  • [1 mark] C′=180(1−4/l²)=0 gives l=2 and b=2.
  • [1 mark] C′<0 for 0<l<2 and C′>0 for l>2, proving the global minimum.
  • [1 mark] Cmin=280+180(2+2)=₹1000.

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