Architectural Designs

A stained glass window for the entrance of a new office building is to be designed in the shape of a rectangle?

capped with a semi-circle. The perimeter of the window is not to exceed 30 m. The price of the stained glass is $9.75/m^2. Determine the total cost of the glass required for this window to have a maximum area. can u please explain what the variables stand for and what equation you are using. Thanks a lot

Public Comments

1. Let x be the width and y be the length of the rectangle. x/2 is the radius of the semicircle Perimeter of the Norman window is x+2y+(π x)/2 Let P be the perimeter P = x+2y+(π x)/2--------(1) Solving for y from equation (1) 2y = P-x-πx/2 y = P/2-x/2-πx/4--------(2) Area = xy + π x^2 / 8 A = x(P/2-x/2-π x/4) + π x^2/8 A= Px/2-x^2 /2 -πx^2/4 +πx^2/8 dA/dx = P/2 -2x/2-2πx /4 +2πx / 8 =0 (4p-8x-2πx)/8=0 4p-2x(π+4)=0 4p=2x(π+4) x= 2P / (4+π) using equation (2) y=P/2-P/(4+π)-2πP/4(4+π) 2(4+π)P-4P-2πP/4(4+π) =4P/4(π+4) = P/(π+4) I have used P for the perimeter. In the last line P=30. Width of the window = 2(30) / (4+π)= 60 / (4+π) =x Length of the window = 30/(π+4) = y Area = xy + π x^2 / 8 Compute the area and multiply by 9.75 to get the total cost. Note: d^2A/dx^2 =-1-π/2+π/4 < 0, indicates that the area is maximized. .
2. 30=2r+2h+pi*r/2 The rectangle must be complete to sustain the semicircle S= r*h +1/8pi r^2 h= (30-2r-pir/2)/2 S= 1/2*(30r-2r^2-1/2pi*r^2) +1/8pi*r^2 S¨=15-2r-pir^/2+1/4 pir = 15-r(2+pi/2-pi/4)=0 so r= 15/(2+pi/4) =5.3852 h=5.3852 Cost = $393.80