Problem 1394

Solution: As shown in Figure 3, let
the water station be $\mathrm{R}(x, y)$
$(0 \leqslant a \leqslant x \leqslant c, \mu \leqslant$ the total cost of the pipeline is
$\lambda \mathrm{TR}+u \mathrm{RP}+$
$v \mathrm{RQ}$, i.e., $\square$
$f(x, y)=\lambda y+u \wedge(x-a)+(b-y)^{\prime}$
$+v,(c-x)^{2}+(d-y)^{2}$.
(If $\lambda \geqslant u+v>0$, the total cost of the pipeline from TP and TQ is
$\lambda \mathrm{TR}+u \mathrm{RP}+v \mathrm{RQ}$
$(u+v) \mathrm{TR}+u \mathrm{RP}+v \mathrm{RQ}$
$=u(\mathrm{TR}+\mathrm{RP})+v(\mathrm{TR}+\mathrm{RQ})$
The above inequality indicates that it is unnecessary to build the main pipeline TR; the water station should be built on the riverbank $x$-axis, at this time $y=0$ and (3) becomes
$\begin{aligned} f(x, 0)= & u \wedge(x-a)+b^{2} \\ & +v,(c-x)^{2}+d^{2} .\end{aligned}$
Solving this problem by referring to the law of refraction of light, where ${ }^{1}=u$,
$1=v$, we have $\qquad$
where $\theta_{1}=\angle \mathrm{PTR}, \theta_{2}=\angle \mathrm{RTQ}$. According to the law of refraction of light, the water station should be built on the riverbank AC, and should be biased towards the side with the higher cost of kilometer pipeline construction.
(2 If $u+v>\lambda>u, v>0$, to use the corollary of Proposition 1 to transform (3) into
$f(x, y)=u \prod_{1}(x-a)+(y-b)^{2}$
$+m(y-b)]+v[1(c-x)+(y-d)$
$+n(y-d)]+u m b+$ und
Here $m$ and $n$ are undetermined constants with absolute values less than 1; they satisfy $\qquad$
(5)
(6)
In (6), it is clear that $m>0, n>0$, calculating $\qquad$
$=\frac{1}{2 \lambda u}\left[v^{2}-(\lambda-u)^{2}\right]$ $\qquad$
Therefore, $0d>0)$, the water station is built at the factory $Q(c$
$d$ location, the total cost of the pipeline is $\mathrm{iCQ}^{\mathrm{CQ}}+u \mathrm{uP}$ $\qquad$
First, if $\lambda=u=v$, the formula simplifies to
Checking condition (1). (1) is ${ }^{1}>^{b}>^{b-d}$ $\qquad$
$=\left(\frac{b}{2}+d+{ }_{2}^{13}(c-a)\right) \lambda$
In the 6th problem, ${ }^{2}+{ }_{2}^{+1}{ }^{3}(c-a)$ is 10 kilometers and 8 kilometers for two factories 14 kilometers apart. Given
$\mathrm{P}(0,10), \mathrm{Q}\left(8,3,8\right)$ satisfies ${ }_{3}^{1}>\begin{array}{l}10-8 \\ 0-8,3\end{array}$
Water station 1 $\qquad$ $\qquad$
The minimum total length of the pipeline is
$\frac{10+8}{2}+{ }_{2}^{1}(8,3-0)=21$