On a suspension bridge the roadway is hung from cables hanging between support towers. the cable of one bridge segment of the parabola y=.1x^2-7x +150 where y is the height of the cable in feet above the roadway at the distance x feet from a support tower. what is the closest the cable comes to the roadway? how far from the support towers does this occur

The parabola opens upward and is symmetric to the y-axis. Its general form is: . y \;=\;ax^2 + c Its y-intercept is (0, 10) . . . Hence: . y \;=\;ax^2 + 10 It passes through (200, 100). We have: . 100 \:=\:a\cdot200^2 + 10 \quad\Rightarrow\quad a \:=\:\frac{9}{4000} Hence: . y \;=\;\tfrac{9}{4000}x^2 + 10 When x = \pm50,\;\;y \:=\:\tfrac{9}{4000}(50^2) + 10 \:=\:\frac{125}{8} Therefore, 50 feet from the center, the cable is 15\tfrac{5}{8} feet high.