Example 5 ・At
7:00 AM, depart from port $A$,
take a motorboat at a constant speed of $v_{1}$ kilometers/hour (4
$\leqslant v_{1} \leqslant 20$) to
head to port $B$ 50 kilometers
away from port $A$, then immediately switch to a steamship at a constant speed of $v_{2}$ kilometers/hour $\left(30 \leqslant v_{2} \leqslant 60\right)$ to travel 300 kilometers to city $C$, and it is stipulated to arrive at city $C$ between 4:00 PM and 9:00 PM. Let the time spent on the motorboat and the steamship be $x, y$ hours, respectively, and try to answer:
(1) Find the range of $x, y$ that satisfies the above conditions,
(2) The travel cost $p$ is given by $p=1000+30(5-x)$ $+20(8-y)$, if the travel cost is to be minimized, what values should $v_{1}, v_{2}$ take?
(1) From the given conditions, we have
\[
\left\{
\begin{array}{l}
x=\frac{300}{v_{2}}, \quad 30 \leqslant v_{2} \leqslant 100, \\
y=\frac{50}{v_{1}}, \quad 4 \leqslant v_{1} \leqslant 20,
\end{array}
\right.
\]
and
\[
9 \leqslant x+y \leqslant 14,
\]
which implies
\[
\left\{
\begin{array}{l}
3 \leqslant x \leqslant 10, \\
2.5 \leqslant y \leqslant 12.5, \\
9 \leqslant x+y \leqslant 14.
\end{array}
\right.
\]
The range of the points in question is shown in the shaded area (including the boundaries) in Figure 6.

(2) From the given conditions,
\[
y=-\frac{3}{2} x + \frac{1}{20}(1310 - p)
\]
To minimize \( p \), the intercept on the \( y \)-axis must be maximized. From Figure 6, it is clear that the line must pass through point \( A(10, 4) \). Substituting \( x=10 \) and \( y=4 \) into (1) gives \( p=930 \). At this point, \( v_{1}=\frac{50}{4}=12.5 \) and \( v_{2}=\frac{300}{10}=30 \).