Question

Solve question number 53

Answer 1

53. Qn.  Maximizing the amount of light passing through a rectangular window with a semicircular arc on the top.

Rectangular part height = a.    Radius of the semicircular part = R
Width of the base = 2 R.

Let the amount of light / unit area transmitted by the colored glass = L  units. This is fixed.  So the amount of light / unit area transmitted by the clear glass is 3 L units.

Perimeter P = fixed = π R + 2 a + 2 R
So a = (P - π R - 2 R)/2

Light Energy transmitted E = πR²/2 * L + 2 a R * 3 L
     E = π R² L /2 + 3 R L (P - π R - 2 R)
         = 3 P L R  - 5/2 * πL R² - 6 L R²

dE/dR = 0 = 3 P L - 5 π L R - 12 L R
=>  R = 3 P /(5π+12)

So  2 a = P - (2+π) 3 P/(5π+12) 
      2 a / P  = (5π+12 - 6 - 3π) / (5π+12)
              a  = (π+3) P /(5π+12)

Ratio of sides :  2R : a = 6 : (π+3)
 
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Qn 52...

Max area of triangle QSR = 4/3√3.

  See picture enclosed.