SOLUTION: a window has the form of a rectangle surrounded by a semi circle if perimeter is 40 units.find its dimensions so that the area of window is maximum

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Question 1041677: a window has the form of a rectangle surrounded by a semi circle if perimeter is 40 units.find its dimensions so that the area of window is maximum
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Assume the window is X wide and Y long.
So then the perimeter of the window would be.
P=X%2B2Y%2B%28pi%2F2%29%2AX
1.%281%2Bpi%2F2%29X%2B2Y=40
and the area of the window would be,
A=XY%2B%28pi%2F2%29%28X%2F2%29%5E2
2.A=XY%2B%28pi%2F8%29X%5E2
From eq. 1,
2Y=40-%281%2Bpi%2F2%29X
Y=20-%28%281%2Bpi%2F2%29%2F2%29X
Y=20-%281%2F2%2Bpi%2F4%29X
Let's just call the constant in front of X, B, just for better readability.
Y=20-BX
So then substituting into 2,
A=X%2820-BX%29%2B%28pi%2F8%29X%5E2
A=20X-BX%5E2%2B%28pi%2F8%29X%5E2
A=%28pi%2F8-B%29X%5E2%2B20X
A=%28pi%2F8-1%2F2-pi%2F4%29X%5E2%2B20X
A=%28-pi%2F8-1%2F2%29X%5E2%2B20X
A=-%28%284%2Bpi%29%2F8%29X%5E2%2B20X
Once again, let's use a constant, C.
A=CX%5E2%2B20X
Convert to vertex form to find the vertex and maximum area.
A=C%28X%5E2%2B%2820%2FC%29X%29
A=C%28X%5E2%2B%2820%2FC%29X%2B%2810%2FC%29%5E2%29-C%2810%2FC%29%5E2
A=C%28X%2B10%2FC%29%5E2-100%2FC
A=-%28%284%2Bpi%29%2F8%29%28X-80%2F%284%2Bpi%29%29%2B800%2F%284%2Bpi%29
So the maximum area of
A%5Bmax%5D=800%2F%284%2Bpi%29
occurs when,
highlight%28X=80%2F%284%2Bpi%29%29
and
Y=20-%281%2F2%2Bpi%2F4%29%2880%2F%284%2Bpi%29%29
Y=20-%28%282%2Bpi%29%2F4%29%2880%2F%284%2Bpi%29%29
Y=20-%28%282%2Bpi%29%2F%284%2Bpi%29%29%2A20
Y=20%281-%282%2Bpi%29%2F%284%2Bpi%29%29
Y=20%28%284%2Bpi%29%2F%284%2Bpi%29-%282%2Bpi%29%2F%284%2Bpi%29%29
Y=20%28%284%2Bpi-2-pi%29%2F%284%2Bpi%29%29
Y=20%282%2F%284%2Bpi%29%29
highlight%28Y=40%2F%284%2Bpi%29%29