A Norman window is constructed by adjoining a semicircle to

Khaleesi Herbert 2021-08-06 Answered
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 22 feet. Radius is x/2.
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Expert Answer

funblogC
Answered 2021-08-07 Author has 91 answers

Let represent the measure of the vertical dimension of the rectangular portion of the window. Let represent the horizontal dimension of the window. is also the diameter of the semi-circular part of the window. The perimeter of the window is then: 22=x+2y+12πx
PTotal=(PRectanglex)+PCircle2
1012x15πx=y
A=xy+π(12x)22
ATotal=ARetangle+ACircle2
A=x(1012x15πx)+π(12x)22
Plug in 1012x15πx for y
A=10x12x2πx210
dAdX=10xπx5
Power rule ddxxn=nxn1
0=10xπx5
Set dAdx equal to 0
x+πx5=10
Add x+πx5 to get x alone
x(1+π5)=10
x=101+π5
x=404+π
d2Adx2=(4+π)4<0 Solve for y:
y=10(2+π)x4
Hence x=404+π is the x-coordinate of the maximum point. The y dimensoin and the semi-circle redius are calculated as above.

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