Question

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.

Answer 1

Consider x as the radius of the semi-circular opening of the window and y as one of the sides of the rectangle.

Write the equation for the perimeter of window.

1 2 ( 2πx )+y+2x+y=10 πx+2x+2y=10 2y=10−πx−2x y= 10−πx−2x 2 (1)

Consider z as the area of the window.

Write the equation for the area of the window.

z= 1 2 π x 2 +2xy

Substitute the value from equation (1)

z= 1 2 π x 2 +2x 10−πx−2x 2 = 1 2 [ π x 2 +20x−2( π+2 ) x 2 ] = 1 2 [ π x 2 +20x−2π x 2 −4 x 2 ] = 1 2 [ −π x 2 +20x−4 x 2 ]

Differentiate both sides of equation,

dz dx = 1 2 [ −2πx−8x+20 ] d 2 z d x 2 = 1 2 [ −2π−8 ] =−( π+4 )

Now,

dz dx =0 1 2 [ −2πx−8x+20 ]=0 −πx−4x+10=0 x= 10 ( 4+π )

At x= 10 ( 4+π ) ,

d 2 z d x 2 =−( π+4 ) <0

Thus, z is maximum at x= 10 ( 4+π ) .

From equation (1),

y= 10−( π+2 )( 10 ( 4+π ) ) 4 = 10π+10−10π−20 2( 4+π ) = 20 2( 4+π ) = 10 4+π

Thus, the radius of the semicircle is 10 4+π and the sides of the rectangle are 10 4+π and 10 4+π respectively.