A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Let
*x *and *y*
be the length and breadth of the rectangular window.

Radius of the semicircular opening

It is given that the perimeter of the window is 10 m.

∴Area
of the window (*A)*
is given by,

Thus, when

Therefore, by second derivative test, the area is the maximum when length.

Hence, the required dimensions of the window to admit maximum light is given by

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