A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at C.A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in

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A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at C.A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in

A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at C.A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in
A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at C.
A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time (see the figure).  She can walk at the rate of 4 mi/h and row a boat at 2 mi/h.For what value of the angle θ shown in the figure will she minimize her travel time?求 θ 度数

A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at C.A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in
连接OB、BC,

∴∠BOC=2∠A=2θ,
∴AB=AC▪cosθ=4cosθ,
弧BC=πAC▪(2θ/2π)=4θ,
设”the woman's travel time“为T(θ),
则T(θ)=AB/2+BC/4=2cosθ+θ,
∴T(θ)的导函数T'(θ)=1-2sinθ,
令T'(θ)=0,sinθ=1/2,θ=π/6,
当θ∈(0,π/6)时,T'(θ)>0,T(θ)单调递增,
当θ∈(π/6,π/2)时,T'(θ)<0,T(θ)单调递减,
T(0)=2,T(π/2)=π/2,

∴T(0)>T(π/2),
∴the woman should walk around the lake from Point A to Ponint C．
故答案为：π/2．
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