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Home » A window is in the form of a 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.
A window is in the form of a 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.
Applications of Derivatives | CBSE | Class 12 | Maths | Miscellaneous | NCERT
A window is in the form of a 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  m be the radius of the semi-circular opening of the window. Then one side of rectangle part of window is  and  m be the other side of rectangle.

Perimeter of window

= Semi-circular arc AB + Length (AD + DC + BC)

 

 

 ……….(i)

Area of window 

= Area of semi-circle + Area of rectangle

= 

= 

 

= 

 and  = 

Now 

 

 

 

 

 

At  [Negative]

 is maximum at 

From eq. (i), 

= 

=  =  m

Therefore, Length of rectangle =  m and Width of rectangle =  m

And Radius of semi-circle =  m

i

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