A church window consisting of a rectangle topped by a semicircle is to have a perimeter p. Find the radius of the semicircle if the area of the window is to be a maximum.
The rectangle has dimensions: "2r" and "x".
Tne semicircle: "\\frac{1}{2}\\cdot 2\\cdot \\pi \\cdot r=\\pi \\cdot r" .
"p=2r+2x+\\pi \\cdot r\\\\\nx=\\frac{1}{2}(p-r(\\pi+2))"
The area of the window is to be a maximum:
"S=x\\cdot 2r+\\frac{1}{2}\\pi \\cdot r^2=\\\\\n=(p-r(\\pi+2))\\cdot r+\\frac{1}{2}\\pi \\cdot r^2=\\\\\n=pr-r^2(\\pi+2)+\\frac{1}{2}\\pi \\cdot r^2\\\\\nS'=p-2r(\\pi+2)+\\pi r=p-\\pi r-4r\\\\\np-\\pi r-4r=0\\\\\nr(\\pi+4)=p\\\\\nr=\\frac{p}{\\pi+4}"
If "r\\in (-\\infty, \\frac{p}{\\pi+4}), S'>0" .
If "r\\in (\\frac{p}{\\pi+4},\\infty), S'<0" .
"S(\\frac{p}{\\pi+4})" - max.
"r=\\frac{p}{\\pi+4}" .
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