如图,现有一块半径为2m,圆心角为90°的扇形铁皮AOB,欲从其中裁剪出一块内接五边形
ONPQR,使点P在AB弧上,点M,N分别在半径OA和OB上,四边形PMON是矩形,点Q在弧AP上,R点在线段AM上,四边形PQRM是直角梯形.现有如下裁剪方案:先使矩形PMON的面积达到最大,在此前提下,再使直角梯形PQRM的面积也达到最大.
(Ⅰ)设∠BOP=θ,当矩形PMON的面积最大时,求θ的值;
(Ⅱ)求按这种裁剪方法的原材料利用率.
考点分析:
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