第144页

信息发布者:
C

0或2或4或 - 2
1
$\frac{4048}{2025}$
70
105
解: (1) 点$B$的坐标为$(3,1)。$
(2) ① 当$x = 2$时,对于$y = -x + 4,$$y=-2 + 4 = 2,$所以$E(2,2);$对于$y = x - 2,$$y=2 - 2 = 0,$所以$F(2,0),$$EF=2-0 = 2。$
当$x = 4$时,对于$y = -x + 4,$$y=-4 + 4 = 0,$所以$E(4,0);$对于$y = x - 2,$$y=4 - 2 = 2,$所以$F(4,2),$$EF=2-0 = 2。$
② 当$x\leq3$时,$y=-x + 4-(x - 2)=-x + 4 - x + 2=-2x + 6;$当$x\gt3$时,$y=x - 2-(-x + 4)=x - 2 + x - 4=2x - 6。$
所以线段$EF$的长$y$与$x$的函数表达式为$y=\begin{cases}-2x + 6(x\leq3)\\2x - 6(x\gt3)\end{cases}。$
③ $k\geq2$或$k\lt - 2。$
解: (2) 设$AB$所在直线的函数表达式为$y = kx + b,$把$\begin{cases}x = 0,y = 210\\x = 3,y = 0\end{cases}$代入得$\begin{cases}b = 210\\3k + b = 0\end{cases},$
将$b = 210$代入$3k + b = 0,$得$3k+210 = 0,$$3k=-210,$$k=-70。$
所以$AB$所在直线的函数表达式为$y=-70x + 210(0\leq x\leq3)。$
因为货车速度为$70km/h,$所以$BC$所在直线的函数表达式为$y = 70(x - 3)=70x - 210(3\lt x\leq5)。$
因为轿车速度为$105km/h,$$\frac{210}{105}=2(h),$所以$D(2,210),$$E(3,210),$$OD$所在直线的函数表达式为$y = 105x(0\leq x\leq2)。$
设$EF$所在直线的函数表达式为$y = mx + n,$把$\begin{cases}x = 3,y = 210\\x = 5,y = 0\end{cases}$代入得$\begin{cases}3m + n = 210\\5m + n = 0\end{cases},$
由$5m + n = 0$减去$3m + n = 210$可得:$(5m + n)-(3m + n)=0 - 210,$$5m + n - 3m - n=-210,$$2m=-210,$$m=-105。$
把$m=-105$代入$3m + n = 210,$得$3\times(-105)+n = 210,$$-315 + n = 210,$$n = 525。$
所以$EF$所在直线的函数表达式为$y=-105x + 525(3\leq x\leq5)。$
由$\begin{cases}y=-70x + 210\\y = 105x\end{cases},$得t$-70x + 210 = 105x,$$105x+70x = 210,$$175x = 210,$$x = 1.2,$$y = 105\times1.2 = 126,$所以$G(1.2,126)。$
由$\begin{cases}y = 70x - 210\\y=-105x + 525\end{cases},$得$70x - 210=-105x + 525,$$70x+105x = 525 + 210,$$175x = 735,$$x = 4.2,$$y=-105\times4.2 + 525=-441+525 = 84,$所以$H(4.2,84)。$
点$G$的实际意义为轿车与货车出发$1.2h$时,在南京与苏州之间,距离苏州$126km$的地方相遇;点$H$的实际意义为轿车与货车出发$4.2h$时,轿车在南京与苏州之间,货车在苏州与上海之间,两车都距离苏州$84km。$
(3) 由题意可知,南京到苏州$210km,$苏州到上海$2\times70 = 140(km)。$