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 horizontal plane where the foot of the mountain is located. Solution: Solution: (1) As shown in the figure, PH⊥α, HB? α,PB⊥AB，

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.