Asked by Joe
A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 39 ft that will admit the greatest possible amount of light. (Round your answers to two decimal places.)
Answers
Answered by
Damon
rectangle height h, width 2r
39 = 2r + 2h + pi r = r(2+pi) +2 h
so
2h = 39 - r(2+pi)
A = area = (1/2) pi r^2+ 2 r h
A = .5 pi r^2 + r[39 -r(2+pi) ]
A = .5 pi r^2 + 39 r - 2r^2 -pi r
A = (.5 pi -2) r^2 +(39-pi)r
dA/dr = 0 for max = 2(.5pi-2) r + (39-pi)
so
r = (39-pi)/(4-pi) etc
39 = 2r + 2h + pi r = r(2+pi) +2 h
so
2h = 39 - r(2+pi)
A = area = (1/2) pi r^2+ 2 r h
A = .5 pi r^2 + r[39 -r(2+pi) ]
A = .5 pi r^2 + 39 r - 2r^2 -pi r
A = (.5 pi -2) r^2 +(39-pi)r
dA/dr = 0 for max = 2(.5pi-2) r + (39-pi)
so
r = (39-pi)/(4-pi) etc
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