Let d represent the diameter of the semicircle (and the width of the window). Then the length of the semicircular arch is π/2*d. The remaining perimeter available for the height of the rectangular shape is then
.. 1000 -(d +π/2*d)
and that amount will be divided between the two sides of the window.
Then the height of the rectangular section is
.. h = (1/2)*(1000 -d(1 +π/2))
.. = 500 -d(1/2 +π/4)
The total area of the window is ...
.. A = dh +(1/2)(π/4)d^2
.. = d(500 -d(1/2 +π/4)) +π/8d^2
.. = 500d +d^2*(π/8 -π/4 -1/2)
.. = d(500 -d(1/2 +π/8))
This downward-opening quadratic function has its vertex (maximum) at
.. d = 500/(1 +π/4) = 2000/(4 +π)
The corresponding rectangle height is
.. h = 500 -500/(1 +π/4)*(1/2 +π/4)
.. = 500(1 - (2 +π)/(4 +π)) = 500(4 +π -2 -π)/(4 +π) = 1000/(4 +π)
That is, the height of the rectangular section is 1/2 the diameter of the circular section, so the overall window has the same overall height and width.
The height and width of the window with maximum area are ...
2000/(4+π) in ≈ 280.0 in
_____
The graph shows both the area function and the (d, h) dimensions of the rectangle.
.. 1000 -(d +π/2*d)
and that amount will be divided between the two sides of the window.
Then the height of the rectangular section is
.. h = (1/2)*(1000 -d(1 +π/2))
.. = 500 -d(1/2 +π/4)
The total area of the window is ...
.. A = dh +(1/2)(π/4)d^2
.. = d(500 -d(1/2 +π/4)) +π/8d^2
.. = 500d +d^2*(π/8 -π/4 -1/2)
.. = d(500 -d(1/2 +π/8))
This downward-opening quadratic function has its vertex (maximum) at
.. d = 500/(1 +π/4) = 2000/(4 +π)
The corresponding rectangle height is
.. h = 500 -500/(1 +π/4)*(1/2 +π/4)
.. = 500(1 - (2 +π)/(4 +π)) = 500(4 +π -2 -π)/(4 +π) = 1000/(4 +π)
That is, the height of the rectangular section is 1/2 the diameter of the circular section, so the overall window has the same overall height and width.
The height and width of the window with maximum area are ...
2000/(4+π) in ≈ 280.0 in
_____
The graph shows both the area function and the (d, h) dimensions of the rectangle.