Traditional Culture Encyclopedia - Tourist attractions - As shown in the figure, in order to develop tourism resources, a certain place wants to build a road connecting scenic spot P and residential area O, and the slope where point P is located and the hor
As shown in the figure, in order to develop tourism resources, a certain place wants to build a road connecting scenic spot P and residential area O, and the slope where point P is located and the hor
According to the inverse theorem of the three perpendicular lines theorem, AB⊥HB,
So ∠PBH is the plane angle of dihedral angle formed by hillside and α,
Then ∠PBH=θ, Pb = PHS in θ = 1.
Let BD=x(km), 0≤x≤ 1.5,
Then PD = x2+Pb2 = x2+1∈ [1,2].
Remember that the total cost is f 1(x) ten thousand yuan,
Set it as f1(x) = (pd2+1+12ad+ao) a = (x2? 12x+ 1 14+3)a =(x? 14)2a+(43 16+3)a
When x = 14, that is, BD = 14 (km), the total cost is minimum.
(2) Let AE=y(km), 0≤y≤54, and the total cost is f2(y) ten thousand yuan.
F2(y)=[PD2+ 1+Y2+3+ 12(32? 14? y)]a=(y2+3? y2)a+43 16a、
Then F2' (y) = (yy2+3? 12)a, from F2 ′ (y) = 0, y = 1.
When y ∈ (0, 1), F2 ′ (y) < 0, and f2(y) is a decreasing function in (0, 1);
When y ∈ (1 54), F2 ′ (y) > 0, and f2(y) is the increasing function in (1 54).
Therefore, when y= 1, that is, AE= 1(km), the total cost f2(y) is the smallest, and the minimum total cost is 6716 million yuan.
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