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NCERT Class 12 · Ch 6 Miscellaneous Exercise · Q8

NCERT Class 12 Maths · Chapter 6 Applications Of Derivatives · Miscellaneous Exercise Question 8

A window is a rectangle surmounted by a semicircle. Its perimeter is 10 m. Find dimensions admitting maximum light.
Maximum-light window: h=r
Maximum-light window: h=r

💡 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: Semicircle radius and rectangle height both 10/(4+π) m; width 20/(4+π) m.
  1. Let semicircle radius be r and rectangle height h. The outer perimeter is 2h+2r+πr=10, so h=5−(2+π)r/2.
  2. Area A=2rh+πr²/2=10r−(2+π/2)r², with 0<r<10/(2+π).
  3. A′=10−(4+π)r=0 gives r=10/(4+π); A″=−(4+π)<0.
  4. Substitution gives h=r. Thus width 2r and total height h+r=2r.

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