Question

A norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. The perimeter of the window is 8m. (a)Write the area of A of the window as a function of r. (b) What dimensions (length and breadth of a rectangle, radius of semicircle will produce a window of maximum area? Write these dimensions in terms of pie and simplify as far as possible.

Answer 1

Perimeter of the window = Length of the arc of semicirle + twice the lenth of the rectangle + breadth of the rectangle

Perimeter of the window = πr + 2x + 2r

πr + 2x + 2r = 8

2x = 8 - πr - 2r

x = (8 - πr - 2r) / 2

(a) Area of the window = Area of semicircle + Area of rectangle

Area of the window = πr^2 / 2 + 2rx

Area of the window = πr^2 / 2 + 2r * (8 - πr - 2r) / 2

Area of the window = πr^2 / 2 + r(8 - πr - 2r) = πr^2 / 2 + 8r - πr^2 - 2r^2

Area of the window = 8r - 2r^2 - πr^2 / 2

(b) Area of the window = ( -2 - π/2 ) r^2 + 8r

Maximum value of a function of the form f(x) = ax^2 + bx + c occurs at x = -b/2a and the maximum value is f(-b/2a).

So as the area function is ( -2 - π/2 ) r^2 + 8r , a = -2 - π/2 and b = 8

Maximum occurs at r = -8 / 2(-2 - π/2) = -8 / (-4 - π) = 8 / (4 + π)

x = (8 - πr - 2r) / 2 = 4 - πr/2 - r

x = 4 - r (π/2 + 1)

x = 4 - (8 / (4 + π) ) (π/2 + 1)

x = 4 - 4π / (4 + π) - 8 / (4 + π)

x = ( 4(4 + π) - 4π - 8 ) / (4 + π)

x = 8 / (4 + π)

Length = 2r

Breadth = x

Length of Rectangle : 16 / (4 + π)

Breadth of Rectangle : 8 / (4 + π)

Radius of Semicircle : 8 / (4 + π)