A window is in the form of a rectangle surmounted by a semi-circle opening. The perimeter of the window is 10 m. Determine the window dimensions that will allow the most light to enter the opening.

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Trinity Walter · Accepted Answer

Let radius of semi-circle = rThus, one side of rectangle = 2r. Let other side = xThus, P = Perimeter = 10 (given)  ⇒  2  x  +  2  r  +      1    2    (  2  π  r  )  =  10  ⇒  2  x  =  10  −  r  (  π  +  2  )  .  .  .  (  i  )Let A be area of the figure, thenA = Area of semi-circle + Area of rectangle   =      1    2    π  r  2  +  2  r  x  ⇒  A  =      1    2    (  π      r    2    )  +  r  [  10  −  r  (  π  +  2  )  ] [Using Eq. (i)]  =      1    2    (  π      r    2    )  +  10  r  −      r    2    π  −  2      r    2    =  10  r  −            π              r        2              2    −  2      r    1  On differentiating twice w.r.t. r, we get            d      A              d      r        =  10  −  π  r  −  4  r  .  .  .  (  i  i  )and                     d        2            A              d              r        2              =  −  π  −  4...  (  i  i  i  )For maxima or minima, put             d      A              d      r        =  0  ⇒  10  −  π  r  −  4  r  =  0  ⇒  10  =  (  4  +  π  )  r  ⇒  r  =      10          4      +      π      On putting   r  =      10          4      +      π       in Eq. (iii), we get                     d        2            A              d              r        2            = negativeTherefore, A has local maximaum when   r  =      10          4      +      π      ...(iv)Radius of semi-circle   =      10          4      +      π      and one side of rectangle   =  2  r  =            2      ×      10              4      +      π        =      20          4      +      π      and one side of rectangle i.e., x from Eq. (i) is given by  x  =      1    2    [  10  −  r  (  π  +  2  )  ]  =      1    2    [  10  −  (      10                  π            +      4        )  (      π    +  2  )  ] (from Eq.(iv))Radius of semi-circle   =      10          4      +              π            and one side of rectangle   =  2  r  =            2      ×      10              4      +              π              =      20          4      +              π            and other side of rectangle i.e., x from Eq.(i) is given by  x  =      1    2    [  10  −  r  (      π    +  2  )  ]  =      1    2    [  10  −  (      10          π      +      4        )  (      π    +  2  )  ][from Eq.(iv)]  =            10              π            +      40      −      10              π            −      20              2      (              π            +      4      )        =      20          2      (              π            +      4      )        =      10                  π            +      4      Light is maximum when area is maximumThen, dimensions of the window arelength   =  2  r  =      20                  π            +      4        ,breadth   =  x  =      10                  π            +      4

A window is in the form of a rectangle surmounted by a semi-circle opening. The perimeter of the window is 10 m. Determine the window dimensions that will allow the most light to enter the opening.

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