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NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q6

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

An open rectangular tank has depth 2 m, volume 8 m³. Base costs ₹70/m² and sides ₹45/m². Find minimum construction cost.
Least-cost tank base: square 2 m × 2 m
Least-cost tank base: square 2 m × 2 m

💡 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: ₹1000; base 2 m × 2 m and depth 2 m.
  1. If base sides are x,y>0, 2xy=8, hence xy=4. The base cost is fixed at 70·4=₹280.
  2. Side area is 2·2(x+y)=4(x+y), costing ₹180(x+y). Since y=4/x, C=280+180(x+4/x).
  3. C′=180(1−4/x²)=0 at x=2; C″=1440/x³>0. The minimum occurs at x=y=2.
  4. Total minimum cost is 280+180·4=₹1000.

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